FF has been a passion of mine in gaming for years! :D
Final Fantasy Unlimited is the series which was pretty awesome. They're worth watching separate to the games because (just like the games) they have no relation to the the other games at all. Just so each game is unrelated (apart from the abomination that is FFX-2) so the series is stand-alone.
I highly advise playing the games however since they're awesome. Best to play 7, 8 and 9 at least (on P.C or Playstation).
FF7 has the most spinoffs in anime. As far as I've seen they all relate to before the game. The best one off I saw was called Final Fantasy 7: Last Order, which is basically the end of Crisis Core. Very impressive stuff.
My advice is get stuck into what has become an entire entertainment medium in its own right. Lose yourself in the Final Fantasy ;)