I have played all ff games on snes, psx and ps2. I have playes but dropped ffxiii and never got into the series again, so I can't really comment on the remake. They are the staple jrpg for me: always good games but without really standing out on anything in particular, except perhaps presentation.
Final fantasy is not so much a series as a collection of unrelated games that share some themes and iterate on the mechanics. They are games about frienship and saving the world from progressively bigger bads. Mechanically they experiment a lot but you can tell they used the systems of previous entries as a base.
Again I don't know if this is still the case for modern entries as they have switch staff quite a bit. In the (offline) mainline games I would recomend you start with the oldest one that does not feel too dated for you. Most of the pre psx entries have remakes too so they are an ok place to start. If you are not really sure you want to dedicate so much time I would at least try ffix and ffx. ffix captures the early nintendo era aesthetic very well and ffx is a fairly well regarded title in general.
That is for mainline titles. I would really recommend the tactics sideline too. The original has a very good story more in line with ogre battle than final fantasy, so lots of medieval political intrigue. The portable one are a bit more childish in tome but very good too.