First: No, the stories do not link. All the persona games have an own story which is closed in the games. Of course there are some characters you might know from older games (For example Igor, who is in every game; But also Chihiro, the student council president of the school of P3 is seen in P4), but you don't need the other ones first.
I started with Persona3Portable. P3P Was a great game, though the overworld was limited in contrary to the not portable versions. Still, it is a great game and I would recommend buying it first so you can see if you like it.
Persona 4 Golden ist atually my favorite game for the PSVita (also almost favorite game in general
). It uses all the functions well and has nice extras, additions to the story and is very roundend. I would really recommend the games for everyone, but you should try P3 first.
Also don't try P2 or P1 to see if you like the others, P3&4 are nothing like the ones before. You might like P2 (I didn't to be honest), but the experience is different from the next games.