My 3 favourite Vita titles are LittleBigPlanet; WipEout 2048 & Gravity Rush. I'd recommend those 3 to everyone.
WipEout can be a little difficult to get into (it's not the most forgiving racer) but it's extremely rewarding. Plus, you can get all the content from HD/Fury as DLC so it's a massive, massive game.
LittleBigPlanet is the definitive version. If you only have a PS2 you won't have been able to play any of the games in this series, so you might as well start with the best. Great story mode and then tonnes of user-created content. Plus, you never know if you might get hooked into making content yourself.
Gravity Rush is just a great game. Not too long, but with a fascinating mechanic that really makes exploring the city (and doing missions) a lot of fun. I didn't want the game to end. Story gets a little iffy here and there though.
I also have Uncharted: Golden Abyss; Need for Speed: Most Wanted; New Little King's Story; PlayStation All-Stars Battle Royale & Super Monkey Ball. SMB is pants & LKS is better on Wii (I presume you have a Wii?) so I can't really recommend those two. Need for Speed is great but probably better on Wii-U (I presume you have a Wii-U?) so that one's out too. Uncharted will be an interesting place for you to get into the series if you've never played one before. It's much more hands-on with gameplay; although the story isn't quite as good. I'd definitely give it a shot.
I don't have that many PSN games, but what I've played of Guacamelee; Hotline Miami & Tales from Space has been fun.
Regarding PS2 games on Vita - there's actually quite a few HD Collections making their way across now. But you'll have to evaluate them on a case-by-case basis. For example, MGS 2 & 3 HD are great games - but if you already played them on PS2 there's no point unless you have a desire to take them on the go with you.
Regarding PS3 games on Vita, there's a few - stuff like Atelier Totori is great (imo) if you like JRPG's and can stomach a little moe. Disgaea 3 is also available on Vita and Disgaea 4 is coming next year. That also brings up that Persona 4 Golden is the highest rated game on Vita - but again, if you played it on PS2, I'm not sure how much more content the Vita version adds.
You could also consider Sly Cooper: Thieves in Time. I don't know if you liked that series on PS2, but the latest game is available on Vita if you don't have a PS3.
There's stuff like Oddworld: Stranger's Wrath HD & Jet Set Radio that are fun games to take around with you everywhere. Earth Defense Force 2017 Portable is also recommended. It was an Xbox360 only game; and it's a fantastic series for some mindless fun.
As for PSP games, well, wow. There's a massive amount of stuff you can try. My favourite PSP games that I've downloaded for Vita so far have been: Dissidia 012 Final Fantasy; Grand Theft Auto Vice City Stories; Ratchet & Clank: Size Matters; Daxter & WipEout Pure. I mean, there's loads of stuff on there. If you like JRPG's - try out Ys Seven or Persona 3 Portable (or Persona 1 & 2). If you like shooters, try Resistance Retribution or Star Wars Battlefront Elite Squadron. If you like innovative stuff, try Patapon.
There's a goldmine on there. For PSP games at least, it might be worth going to the Sony Entertainment Network store:
https://store.sonyentertainmentnetwork.com/#!/en-gb/home/games
Browsing through the PSP games section and see what takes your fancy.
Finally, don't forget there's a massive library of PS1 games available on Vita too. I know you said that you already own a PS2, so they might not interest you that much, but I've definitely enjoyed being able to take Spyro the Dragon & Metal Gear Solid 1 on the go with me.
