They can be played individually. The only direct connection between games is when it's sequels and prequels, but those tend to be obvious, like FFX and FFX-2, or FFVII and Crisis Core FFVII, stuff like that. Otherwise, they are individual games whose connection with one another means diving in a hole so deep not even hardcore fans know where it's at right now.

They have some reoccurring characters but that's about it and they don't really tie into each other in most cases.

One example being Cid, who is usually tied with getting some form of transportation. You'll often see the same summons and equipment too. FF9 has some Easter eggs referring to previous games.

The only game I can think of that has different characters from different games interacting with each other is Dissidia.