No you dont. Each game has it's own story, characters, world, except the direct sequels. So of course if you plan to play FF 13-2, play 13 first then the sequel (and then 13-3 😛) This is only an issue for 10 and 13.
Also FF 14 is an MMO (quite good actually) so you will write your own story there!
Just for teasing you should take a look at the newly released FF 15 trailers!