You won't need to know anything about the plot or past mysteries to jump in. Basically you will just need to know who the main characters are who are still on the show and how they relate to Veronica and each other. Warning: Contains SPOILERS for season two.
Veronica, is the main character. Two years ago, she became involved in private investigation as a way to come to terms with the then-unsolved murder of her best friend, Lilly. She lives with her dad, Keith, and is a freshman at Hearst College. Last year she investigated a series of rapes on Hearst's campus. She never found the rapist.
Her best friend Wallace is also going to Hearst.
Her boyfriend, Logan, saved her life on graduation night when she was lured to the roof of a hotel by a psychotic student named Cassidy (nicknamed "Beaver"). At the time, she and Logan were 'formerly dating' and not on great terms, but they went through a lot together at the end of the season, and their love for each other was more important than the bullshit they put each other through and they got back together.
Mac is Veronica's only female friend. (She had two others, but they are dead, so you don't really need to know about them.) Mac was dating Cassidy when he tried to kill Veronica and then killed himself, so she is pretty traumatized right now. She is a computer whiz, and the "Q" to Veronica's "Bond." (Or the Marshall to her Sydney.)
Dick is Logan's best friend and Cassidy's older brother. He just lost his brother (who he found out was molested as a child and behind the murders of 10 innocent people), so he is pretty messed up at the beginning of the season. To be fair, he was already pretty messed up. He is simple kid with a lot of money and not lot of conscience or consciousness. Like Veronica says, "Dick is... Dick." Up until now, he has pretty much served as comic relief.
Keith, Veronica's dad, used to be the town sheriff. Now he is a private investigator. It is through him that Veronica has learned all she knows about the business. For the past two years she has been his apprentice, sometimes with his permission, sometimes behind his back. He is single now, but it is pretty clear that even when her mother was in the picture, V was always her daddy's girl. They are a lot alike and sometimes clash because of it, but mostly they have a very healthy, if offbeat, parent/child relationship. Kind of like the personality of Lorelai and Rory Gilmore with the circumstances of Jack and Sydney Bristow.
Kendall, a secondary character, was married to Dick and Cassidy's father and involved in a lucrative real estate scam with Cassidy, leading her to inherit a bunch of money when he committed suicide. At the end of the season finale, she came to Keith with a "business proposition" involving the mysterious contents of a briefcase. We don't know what she asked Keith to do for her (or what was in the briefcase), but whatever it was, it caused Keith to abandon Veronica at the airport. They were supposed to go to NYC together as her graduation present.
The creator of the series has already said that the first episode of season three will serve as a "second pilot" for the series, so you shouldn't even worry about anything I have just told you. The writers will tell you what you need to know to catch up. But it never hurts to be familiar with the cast of characters. Shakespeare does it.