There are some minor references to the earlier games (one of which may actually connect it to the main Persona timeline) but as Karellen said story wise they are not directly connected to each other. Though I still suggest at least playing P2: Eternal Punishment as it is a fine game as well.

P3 even plays differently from the previous installments, its combat system is a more merciful version of Nocturne's (exploiting elelmental weaknesses to get extra turns while trying to prevent the enemy from doing the same.). There is a high level of strategy involved but it doesn't have Nocturne's brutal sometimes unfair difficulty, causing you frustration.

P3 is one part RPG dungeon crawler, one part dating sim... Its an odd combo but it works out beautifully and the myriad of stories you unlock through the Social Link system are not only touching and downright amusing but create a better immersion to the game world. The games an odd premise but its execution is damn near flawless.