Spoilers for Persona 4, and just warnings in general for repetitiveness and tl;dr and incoherency.
1. The Lovers Arcana
"So when I was alone, I was truly alone. Whatever I was thinking about... I had to keep them to myself. So... right now... I don't want to be alone."
In all honesty, I can't write an essay about Rise and relationships without mentioning her arcana-- because there's a reason why she's the Lovers arcana, and it's not just because she ships things like a Titanic that stays afloat. Strong, working relationships have a deep form of love, not just romantic, but a bond nonetheless. It's a card about love, and it's a card about choices.
The card also represents a decision, the farewell to a former way of life and the full acceptance of love uncompromising. It doesn't necessarily mean that the 'love' is a person - it can be a passion, a desire, or even a profession. Many people who lead an adapted, settled life actually have deep desires and hidden wishes, that they don't dare show, that they constantly fight back, suppressing their own luck.
The Lovers can mean that someone should be honest, make a clear decision for this love, and stand by it.
Rise's decisions in regards to her lifestyle are all based on love. It's not the easiest thing to completely change yourself and break out of shyness like you just took a sledgehammer to a fishbowl, but she does manage. Why? Because she wants genuine relationships. Why does she quit? Because no one knows about that single desire she's been suppressing, because she's being viewed as something she's not, and because people love her for entirely the wrong reasons. Why does she accept - and apologize - to her Shadow? Because her Shadow represents Rise's desire to be looked at, really looked at, and accepted for who she is, not who she's been told to act like as if that's all there is to her. Why does she return to showbiz? Because she wants to show the world all of her, promising that she'll always be herself, and if her audience can love her, as flawed and imperfect as she is, that's the ideal moment for her.
I'll also mention that she's got no reservations whatsoever in informing people that she loves them, both because it's a nice, lighthearted thing to do, and also because it's such a relief to be able to say it and mean it.
2. Rise and friendship
"I wasn't interested in being an idol, but... I wanted the chance to change myself. And just by chance, I won. Haha... when it happened, I panicked, thinking maybe I should turn their offer down. Then I thought... if I appeared on TV, I could make friends... That was literally the only reason I went through with it..."
If you know Rise, you've probably realized by now that her friends are the most important things to her. But why? Sure, friends are great, but so is money. So is education (shut up, it is to someone). Her background plays a big part in this answer, actually, so let's get into that.
She's an only child and therefore never had that sibling bond with anyone. She does play the energetic and affectionate older sister to Nanako, but she's the team's baby sis, anyway.
Her parents are strict, which does tend to put a damper on friend bonding. Rise explains in the game that she's always wanted to eat out with friends, for the atmosphere, but that her parents wouldn't allow it. Her friends as a kid were few and far between.
She was bullied-- maybe because she was shy and looked at her feet when she spoke, maybe because she was pretty, maybe both, or a different reason entirely, kids are kids. But it doesn't change the fact that right up until her debut (which happened in middle school), Rise was the kid getting shoved down on the playground, the one opening her locker to find her shoes filled with tacks or covered in mud, the one getting threatened behind the school, you get the idea. As an idol, she later runs an public service campaign against bullying (which I do believe was entirely Rise's idea, as it's the one thing she seems adamant about at the time), and she still doesn't tolerate it. She can't stand bullies and she won't bully, because she understands what it's like to be that victim.
Her reason for becoming an idol has already been mentioned, but that's Rise's sole reason. It wasn't a dream, not for someone who claimed she didn't mind being alone, because it meant no one expected anything out of her. But her desire for bonds, for that kind of relationship with somebody prevailed over her own sense of discomfort and fear of disappointing anyone by not meeting expectations. She made the decision to change herself because she thought that would garner her some friends. Of course, this backfired, because her idol self is only one part of Rise, but it was also only that part that everyone was interested in befriending.
Rise knows that she, herself, as a whole person, is nowhere near as glamorous as Risette. Her dungeon, where you hear her answering interview questions, is proof of that, though you don't quite realize that the answers Rise is giving at the time aren't the true ones. It's what was expected, what people wanted to hear, so instead of "I only did this so I'd make friends and boy did that go horribly", it was "Of course, being an idol's always been my dream!" She questions herself and the people who know her, asking if they prefer the idol over the girl from Inaba before she's able to see herself as both, but she still has that worry. There's a reason why she doesn't always introduce herself as an idol from the start, unless the conversation's geared in that direction. She doesn't want that fogging over people's views of her, because if she's going to make friends, she wants to make them as Rise and not due to her status.
She will never, ever take a friendship for granted. She didn't have true friends until the Investigation Team, and in Route, every friendship she forms is something she will fight like hell to keep-- even with Hayate and the acquaintanceship turned tentative friendship that kind of ended horribly for awhile until Rise persisted. She yelled at Fuji exactly because she wanted to be his friend, and the circumstances weren't the best, but she wanted him to know that she cared. There's a reason why she's so friendly from the get-go, especially if she recognizes qualities of her shy self in their mannerisms. She can't leave people alone, nor will she, as long as she feels that there's something she can do.
Of course, there are problems she has with it on occasion. If something does happen to a friend (especially to a sibling figure, like Heather, Lightning, Yayoi, Akaya, Marona, Otacon, etc.), Rise will get involved as best as she can. This is sometimes easier said than done, because it can range from stuff like going with the Keyblade Trio to break Vanitas out of Rocket headquarters to running back on a sprained ankle to help Minnie to sticking her nose where it doesn't really belong (i.e. Niou and Yuushi's "little spat") to taking burdens onto her own shoulders. Rise's got the mentality to handle it and, as her own manager says, "she's strong enough to use her pain as a foundation", but the thing is, she's still only one girl and she cares far too deeply for her friends so that when she can't help them, it's agony. It's also frustrating, so that she'll get mad when it's stuff like Souji telling them to stay put, because dammit he's their leader and they're his team and they want to be there for him. Likewise, if any of her teammates try to act as though they're fine, and she knows better, she'll most definitely get upset.
When Izanami's aiming for Souji and her friends are sacrificing themselves for him and being dragged down to their deaths, one by one, Rise is the only one left standing. Screaming their names, begging for them to get back up, because she's the only one of the group who can't fight, and if nothing else solidified it for her - not including Nanako suffering in the hospital, of course - it was that helplessness that drove it home. She hates not being able to do anything for her friends. If you want to cripple Rise, you don't do it by breaking bones. You do it by hurting the people she cares most about and ensuring that she can't do a damned thing to help them. You make her watch.
She may not admit this all that readily, but once Rise has those bonds, she's both so very grateful for them, that people genuinely like her for being who she is, and also so very scared because that's another person she wants to look out for and cherish and sometimes she just... can't. She's terrified of failing anyone who ever thought she was important enough to give a damn about.
Admittedly, not everyone feels the same way she does, and she gets that, and not everyone wants to be befriended by an optimistic scanner idol, but once she declares you a friend, you're pretty much stuck with her and all the teasing, harassing, and caring that goes with that. Sorry, Santa.
3. Trust levels
"I'm sorry... I've practically lost touch with what the normal me is like."
There's a reason why Rise's CR chart is done in the manner of a Social Link, and not just because I hate myself. It's also because Rise has divisions of friendships within the grand scale of ~*FRIENDSHIP*~. Not that she won't risk her life for all of you, because of course she will, but there are some things she knows she can trust people with and there are some things she's still wary when it comes to talking about them. So I'll phrase it like this:
Ranks 1-3: You may know the basics about this girl. My, she certainly seems friendly and approachable and GOSH THAT OPTIMISM.
Ranks 4-5: You're learning more, she's confiding in you a little bit more, maybe you're confiding in her a little bit more, you've seen her sad, and maybe she's tried to hide it, but maybe that's one thing she hasn't. Still, there's more to her than just this, you know it at this point.
Ranks 6-7: YOUR INNER DARKNESS TOOK THE FORM OF A WHAT AND YOU DID WHAT BACK HOME JESUS CHRIST-- oh are we bros/sises/brosises/close now, oh okay--
Ranks 8-10: There's pretty much nothing you don't know about her.
It's absurdly easy to get into the first three ranks, because Rise tends to friendly at nearly everyone. Ranks 4 and 5 mean either she's decided to start revealing more or that something's happened and you've learned more regardless and you still feel like happy fun bonding times. Getting above that is hard, and it's not just a matter of how long you've known her or how often you hang out. Sometimes Rise can't bring the topic up, or sees no need to. It took Kaito forever to find out, and even then, it was out of her hands, like with the Shadow fiasco last November. With Fuji, Rise told him everything because she'd been very lightly hinting at it over the months to test it out, and then he pushed her too far at one point and the rest was history. Yayoi was both unintentional and Rise's choice. Yuushi found out, but Rise wanted to trust him, while she and Santa are dancing the awkward tango of WE DON'T REALLY WANT TO DISCUSS WHAT WENT DOWN AT ALL.
See, the thing is, Rise's trust levels are tricky. She can care about people very deeply, but when it comes to talking about the Personas and what she went through, she always encounters a high, nearly insurmountable wall of fear. Bringing up the incident with Hayate again, when he was scared of her because of subconscious Shadow memories - and the same with Yayoi - one of Rise's biggest fears is that once her friends find out about that, they'll backpedal. She can't blame them, because she was rather horrified herself, but it does hurt and she selfishly doesn't want to lose anyone because of her own shortcomings. She doesn't want people thinking she's some kind of depraved, attention-starved stripper, and people do tend to get the wrong idea from that thing. There's explaining it to someone who comes from a world with Personas or magic versus someone who comes from a normal world without it being in danger. There's explaining it to someone who actually saw her Persona or her Shadow in action versus explaining it to someone who may possibly think she's crazy. She hesitated on people like Fuji and Kaito because she couldn't pinpoint what category they'd fall under, while people like Merlin and Ven had an easier time understanding. Santa may have seen what happened, but as neither of them have mentioned it, they're at the point where they know the other has secrets, but they're not comfortable discussing them, so they fall back into that cynicism vs. optimism banter they tend to do.
But the other side of the issue is that, while she's scared, she also feels guilty for not telling her friends everything. She'll always have to watch herself in case she slips up-- and the closer she is to someone, the greater the chance increases that she will, that she'll lose her grip on what she's hiding. You know, "Hi, this is me, you may have to handle all of this and it's the kind of thing that might make you run away screaming--"
Of course, she still acts, and again, it's more likely to occur around people she's not as close to, because for Rise, she can smile or tilt her head or gain a certain expression like it's nothing, and if you don't know her, then yes-- you'll be fooled. She's been trying not to do this as much, but it's still very much a coping mechanism for her, and she can't rid herself of the habit all that easily. Sometimes it's easy to tell: Crocodile Tears vs. Real Tears, for example. Very often, you'll have to call her out on it, unless you're her teammates.
Otherwise, you can usually tell where you're at with her, trust level-wise, by how many sides of her you see. If it's just the playful or pouty side, she's being cautious. If it's playful, pouty, angry, serious, content, that's more Rise. You can get her complaining about homework, but you can also get her suddenly saying something insightful about human nature, or abandoning her usual bouncy and effervescent energy for a more quiet side. She's a good actress, but when she knows you, she wears her heart on her sleeve.
And if she trusts you and you take advantage of that for the purpose of hurting her or anyone close to her, good luck ever getting that back.
4. IF YOU WANNA BE MY LOVER--
"Ahh, I'm scared! I'm scared of putting the real me out there for people to see... But... I can't give in to that fear, right?"
Rise is a teenage girl and, as such, can and will appreciate good-looking people. There's a difference between crushes for her and deeper feelings, and again, you sort of have to know her in order to be able to tell.
She flirts, and she flirts with guys who will either fluster and flail at her or who will flirt back and understand that it's not serious for either of them. It's Rise's method of teasing. If she is genuinely interested in you, it means you'll start affecting her. Even if you go along with the flirting, Rise will get flustered and uncertain and most definitely nervous, because she's starting to wonder if you mean it. She's confessed her love in dramas, never in real life, and when it comes to not acting, she's painfully shy about doing it. Because here's the thing-- she's kind of had bad luck in the romance department. First there was Souji, then there was Ven, and it does tend to get a little discouraging.
You could argue that Rise lives vicariously through other people's love lives and that's why she's always encouraging and pushing and facilitating adorable and romantic interactions between her friends. Maybe because she doesn't think it's too likely for her, being an idol with a schedule like a jigsaw puzzle and the paparazzi and stalkers to deal with, or maybe it's just another method of looking out for her friends. When she still had that crush on Souji, it was both to help a friend and to get her some more time alone with him, but she isn't at the point right now with anyone where she'll make a bias known.
... She is forever going to platonically love her leader, though, so don't expect her to ever not fangirl him.
If you think being close friends with Rise is tough stuff, getting into a romantic relationship means working even harder. First, she's so used to fans and stalkers and impromptu confessions that she won't take them seriously (sorry, Yosuke). Secondly, she's got some insecurities beneath that smile. She needs to know that you're more than okay with her being a tofu-savvy girl who prefers traditional sweets and spicy food, who will pout and sulk if she feels like she's being neglected, who will snap at you if you mock or insult her-- she needs to know that you like her for being all that, and not the ditzy idol cliché, or the girl whose agency had her make her bust size two cm larger in her profile. She's basically been told that she (as herself) is inadequate, and to be loved, she needed to act a certain way and look a certain way, and that's not what she wants in a relationship. She needs someone she can be herself with, and let her guard completely down around, and they won't think ill of her for it. And you know, someone who doesn't just stand there and watches her cry.
Of course, she's even more scared of reaching this point with someone after everything that's happened. Souji at least saw Rise's inner darkness for himself, Ven heard of it early on, and she is not even going to consider seriously dating anyone until they know all of that and are somehow able to accept her for it.
Rise's fine with non-serious dates, fun shenanigans with someone else, going out dancing, but when it comes to asking her to go steady, you sure as hell had better be serious. If you're asking her because she's cute and she just might put out early on, she's gone like that. Don't make her into a sexual object; she won't stick around if you do.
tl;dr: She comes with a warning label and 3/4 is in fine print. Rise knows what it takes to be a friend, but this is new and scary territory, so be gentle with her.