Um. So I set out to put my understanding of Tavros' relationships with his fellow trolls down on paper since I'm planning on trying to RP him, and it got really, really long. Stupidly long.
So, ok. Let's do this from the top of the hemospectrum to the bottom.
Feferi: He has never talked to her in canon! I think that he is probably intimidated by her and by what the other highbloods would think if he went out of his way to speak to her, but if they ever did have a chance to talk they would get along very well. I think that her energetic personality would -EXCIT----E him, and although she is mischievous, she isn't dangerous or hostile like many of the others.
Eridan: Probably never talks to Tavros is he has a choice. Tavros has the dubious honour of being one of the few people Eridan hasn't hit on, so I think he's too low for the seatroll to dirty his hands with. Possibly he jealously stares Tavros down if he ever so much as looks in Feferi's general direction.
Gamzee: Probably the person Tavros feels most comfortable with, because Gamzee is the least judgmental person ever. So Tavros can open up to him and not worry about looking stupid, or offending him, or anything like that. Because of this, he is Tavros' best friend after Aradia dies... but Tavros isn't his. }:( Unfortunately, in a way Gamzee's friendship is so cheap that it doesn't count. He is everyone's friend to the same degree, always willing to offer advice in a useless, spacey way. But it doesn't matter to Tavros that his advice is terrible and that his interests are inscrutable as long as they can rap together and he doesn't have to worry about any hostility. Tavros needs this relationship, I think, even though one side of it might not have a lot of real substance. He has a lot of respect for Gamzee. Maybe red feelings, if you ship that way.
Equius: He probably never spoke to Equius before the Veil. He'd never initiate that conversation, at least, although Equius might have objected to his friendship with Nepeta. Later on, I imagine he is ridiculously thankful for the legs and goes out of his way to awkwardly thank the guy over and over, which possibly the blueblood can appreciate since it's Tavros showing proper respect to a superior.
Vriska: Um. I imagine he didn't know her very well before the accident. If he did, he would know about the whole "man-eating spider" issue. Post-accident, he wants nothing to do with her. At all. Him telling her that Rufio didn't want them to talk kind of implies that she kept in touch with him in the time between the accident and SGRUB. He has no idea why she'd talk to him, but I imagine him trying to give extremely booooooooring answers to try to ward her off without her figuring it out. And she figures it out every time.
During SGRUB, I think a big scene between them is when she tries to make him apologise for being handicapped, and he tells her no, that's unreasonable and I don't want to, and he doesn't back down even though he is sad and alone. This is important! He doesn't do everything she says and even at this point he isn't a coward or deathly afraid of her.
With some prompting from Kanaya, she helps him achieve his lifelong dream of flying. He gains an army of monsters and solves puzzles and it's so awesome wow and Vriska isn't making him feel terrible or anything! He is having fun and he is mobile and he even lets some of his dorky language slip ("strict progress"), which mostly only happens with Gamzee. Then Vriska gets bored because let's be honest, finding segments of a puzzle when they're scattered throughout a desert that is all the same is like playing Diablo II all over again goddamn it Act II pretty dull. Anyway, whoops, fun is over, and the strict progress gets tucked away. He doesn't have any choice but to do what she tells him once more.
And whoa, suddenly he's in her hive and she's dressed as a fairy and oh god what limbs will she ruin this time??? Now that they're face-to-face and she's holding him up, the genuine terror kicks in. And then lots of confusion. He is almost immediately worried that he upset her, and he apologises in the wimpiest way using the memo. But Vriska is a big girl, she gets over it and they go back to having a good time! They are finding treasure and zipping through the air and oh look, now his former friend is murdering the person who he's been having fun playing with. GENUINE DISTRESS!!! Since I ship ridiculous numbers of things, it's not impossible that he started developing red feelings for her in the period leading up to this. Or not, whatever. Point is, he was enjoying playing with Vriska until Aradia popped up and ended that.
And then he failed to kill Vriska and retreated to the dream world for more-or-less the rest of SGRUB. He failed her big test for him and let her bleed out painfully, and he can't deal with it so he proceeds to basically avoid her as much as possible. Her dreamself is gone, after all, so Prospit is safe for him.
The next time they speak, they're in the Veil and Tavros finds out about her involvement in the creation of Bec Noir. He tells her that he's going to try and stop her, getting over both his fear of her and his escapist habits, but too late and in the wrong way. Possibly she would have let him live it if it was a caliginous thing, but he never hated her even at that point.
Post-death, I have a hard time thinking he holds it against her. His quality of life is vastly improved by it, so to speak. He has successfully absconded from the unpleasant sequence of events that made up his life and can hang out with people who he respects and likes (Jade, Dave, Aradia).
Wow that was outrageously long. Jeez, Vriska.
Terezi: His only interaction with her is through FLARPING. Presumably he knows her as poorly as he does Vriska, but he does turn to her for help after Aradia fails to respond. I think that he probably knows about his involvement in her blinding, but once again he avoids the issue. It's noteworthy that she did ask Aradia how he was doing after his accident, so she may keep a closer eye on him than he knows.
Kanaya: Although they never speak onscreen, Kanaya offers him advice for improving his self-esteem pre-SGRUB, and twice he wants to solicit her help for dealing with Vriska, once successfully. The other time, he doesn't get the chance. I interpret their relationship as being like one between a teacher and pupil. He turns to her for advice, but he wouldn't goof around with her or really spill his guts like I could see him doing with Gamzee or Nepeta. She is probably the driving force behind him getting new legs from Equius, which he is incredibly grateful for even though it was kind of upsetting having his legs chainsawed off.
Nepeta: A friend! A real, completely sober person who has no reason to feel guilt for his condition, which means she may be his most genuine friendship since Aradia's death. He is a pretty awkward roleplayer, but I see it as a regular thing that the two of them do, even throughout SGRUB and in the Veil. He cares for her safety and trusts Equius' judgment on the matter.
Karkat: "HEY ASSHOLE, STOP PLAYING GAMES FOR GIRLS."
Tavros thinks Karkat is mean and intimidating, but he trusts him as a leader and as a source of advice, kind of like a Kanaya who yells more. He thinks Karkat dislikes him, so he avoids him unless something important is happening. I wonder how many times I can say "avoid" in this analysis.
Sollux: Is unpredictable and sometimes worse than Karkat. The two of them have so little in common that they probably never talk ever.
Aradia: The best friend Tavros ever had. His partner in Team Charge and one of the few people in the comic to say genuinely kind things to him and to appreciate and encourage his dorky, fanciful ideals. "y0u might be the 0nly flarper in the w0rld wh0 really understands the true spirit 0f the game"
He didn't know why (Aradia asks Nepeta to keep either her death, or her integration with Frogsprite a secret in 4093. I assume it is her death), but his closest friend suddenly dropped all of her interests and had a personality reversal. Suddenly she was spooky and depressing, and much less talkative. I imagine eventually he stopped trolling her.
The next time he sees her, she beats Vriska to death before his eyes,was scarring him forever. Any desire he might have had to talk to her vanished at that point. She is a major contributor to the situation he is trying not to think about.
After his death, Aradia is back to normal and Team Charge rides again! }:D yEAAAAAHHHHHHHH! Sure, she's busy flitting around dream bubbles and expositing, but who cares!? She is herself again.