you can learn a lot by reading this article here: http://en.wikipedia.org/wiki/Fabula_Nova_Crystallis_Final_Fantasy_XIII , and by reading the articles for the 3 games that are linked from that article.
but in a jist, ill attempt to answer your quesitons.
none of the FF games, like others have said, are linked to each other, except for FFX and FFX-2. the most popular of the series, however, tend to be FFVII and FFX, which are both very RPGish. most of the FF games[i havent played any before FFVII, so i cant speak for them] are turn-based, so little-to-no quick thinking is required. all of them have unique battle systems though, with FFXII being the only real-time battle system. you can surely play FFXIII without playing any other FF game, but i highly recommend investing in a few of the previous FF games. you can also play it without playing either of the other 2 games in the Crystallis series, as they all 3 take place in the same universe but are stand-alone from one another. the FF used to be exclusive to the playstation brand, but last years E3 that changed when FFXIII was announced to be multiplatform. FFvsXIII is still exclusive to the PS3[as of now], and FFXIII agito is exclusive to the PSP.
