How about starting with P3? Depending on which platform do you have you can try either P3FES on PS2 or P3P on PSP/PSV. I have played all 5 games so far and I would say persona 2: sin and persona 2: punishment have the best story but they are far away from the modern persona games while more similar to the shin migami tensei games. Persona 3 has pretty good plot and settings, you also get accompanies like A.I robot, dogs, and the idea of using a fake gun to trigger your persona is way cooler than the glasses in 4 and the masks in the upcoming 5. Also need to mention that in P3P you even get the chance to play as the female protagonist!
