Some of the newer, current versions of FF include things mentioned from some of the earlier types of FF (i.e. the mention of crystals), they're related but it's not like you have to play them one by one to understand what's going on.
There's one thing though. Last I checked (don't attack me for this but i'm still a true fan ;-;), some of the FF games were classified into different types of FF themes such as Fabula Nova Crystallis which includes FFXIII, FFXIII-2, FFXIII Lightning Returns, Type-0, & FF Versus XIII (if it's ever released under that title at least).
There's one thing though. Last I checked (don't attack me for this but i'm still a true fan ;-;), some of the FF games were classified into different types of FF themes such as Fabula Nova Crystallis which includes FFXIII, FFXIII-2, FFXIII Lightning Returns, Type-0, & FF Versus XIII (if it's ever released under that title at least).