All Final Fantasy games have separate & unrelated stories. The only true sequel to come out (so far) is FF X-2 wich (obviously) is the sequel to FFX. Oh yeah, there is a sequel to FFVII coming out, but it's not an RPG, it's an action adventure.
So, needless to say, there is no need to play the games in order, unless you wanna see how the game's features & technology have progressed over time.
I recomend playing through all of the FF games if you have the time and/or you can get yor hands on all the games. All the FF games are a blast to play through. If you don't have the time I recommend you play FFVII first, it's the best one in my opinion.
Dip-Set all day, everyday!!