Like the people who are posting more recently in this thread, I'd say go with a "classic" before you go for something like Persona 4. I enjoyed P4 a lot, but the fusion and one more stuff+time management may be a little daunting for a newb to JRPGs, plus it's definitely a more anime choice than most.
Chrono Trigger really is a safe choice for first RPGs. It's shorter than many others, it's easy enough (but not TOO easy especially for a first game, I think it would be a bit of a disservice for your first game in a genre to not give you a little resistance), and the systems that the game presents are easy to understand. Plus it's a solid game in its own right.
For Final Fantasy, IV-X work well, although perhaps not V or VIII (V possibly being too tough, and VIII possibly being too convoluted). I'd personally say IV, IX, or if you want something more modern, X.
Dragon Quest games make wonderful choices because they're uncomplicated, has just the right amount of difficulty to be approachable to many, but are still VERY charming and gives you that feeling of adventure that people often really love about the genre. Unfortunately, with the systems that you've listed, your options are limited, but Dragon Quest 8 is not a bad choice at all. The world in the game is gorgeous and refreshing even today, and the party members, and characters you meet are fun. Like most Dragon Quest games, the story is uncomplicated, but it's more about the things you see along the way instead of the journey itself.
EDIT: Having said that:
I would recommend against DQ8 for two reasons: 1) the very beginning of the game is unusually hard compared to many other RPGs, particularly other Dragon Quest games. I died to the first boss in the cave on my first try, and I was no amateur RPG player at that point. The boss on the ship can also be a surprisingly tough road block. If you are used to RPGs and know how to use your abilities and resources well, it is a great game, but for a true beginner, I think the start can be hard.
2) It is really easy to ruin your party if you don't know what you are doing with your ability point distribution, and there is no way to fix it without restarting the game. It takes a certain degree of forewarning to how many points you will actually get and what the benefits of the various skill trees are to really enjoy the system, I think. In comparison, Dragon Quest IX is more beginner friendly, I believe.
Thinking again on my experience with DQ8, I have to admit that it was similar to yours, especially point 1. I also wasn't a newcomer to JRPGs when I got to DQ8, but that first section was kind of tough, and the ship boss kicked my ass a few times. It would have been nice if the game let you use Jessica before the ship instead of right after so you at least had another body to soak up hits, but that's not the case. I'm still hesitant to unrecommend DQ8 though because the games are essentially the quintessential RPGs, but most of the DQ games nowadays are found on Nintendo consoles, except DQV (which would require a certain amount of effort to play on a Sony console) and DQVII (which also isn't the easiest game in the world to obtain).
For Action RPGs, you really can't go wrong with Ys. Oath in Felghana is a blast to play, short, and the difficulty settings gives it some nice replay value.
Tales of the Abyss/Graces/Symphonia etc. is worth considering if you want a more traditional style JRPG story but with an action battle system. Ys games are fun, but not story or even arguably character heavy, and Tales usually places a more equal balance on all fronts. It is more anime than most of my choices though, so pick carefully if you go with that route.
While there's other great "My 1st JRPG"s on other systems (DQV, Earthbound, and Mother 3 stand out to me), there's absolutely no need for you to buy another system just to try those out unless you see something that looks awesome to you, especially since you have already have a PS2. PS1 and PS2 were monsters when it came to these sorts of games, so you should have plenty to choose from both on that and for your Vita.