I don’t own a Vita but I can recommend( I don’t know how good the vita versions are) some games from its library that I’ve played on other consoles.
Borderlands 2 a shoot and loot rpg fps which I've heard is not the best port ( they had to reduce the co-op from 4 to 2 players and resolution,frame rate etc.) But still enjoyable.
FFX/X-2 Very good jrpg's.
Muramasa: The Demon Blade Really enjoyable side scrolling platforming beat em up rpg.
Dragon's Crown side scrolling 4 player beat em up rpg made by the creators of Muramasa: The Demon Blade. Best with co-op but still fun without.
Tales of Hearts R & Tales of Innocence R they’ve changed Tales of Hearts from 2D to 3D for Tales of Hearts R so I don’t know which is the better version but I find the tales of games have some of the best real-time jrpg gameplay out there so I’m sure their very good.
On the PSX and PSP side I’d also recommend Final Fantasy 1-9 and other Final Fantasy titles like Dissidia Duodecim ( Note: unfortunately Final Fantasy 7:Crisis Core and kingdom hearts birth by sleep which were two of my favourite psp games never made it to the PSN store).