All the mainline games are not interconnected at all, they are pretty much very separate in terms of story / settings / characters. So you can jump into any one of the games. Also, their turn-based systems, aka Active Time Battle, aren't anything interesting, compared to say Shin Megami Tensei's Press Turn system. All FF games have very linear / streamlined experience compared to other games, i.e. choices don't matter much, you don't choose the stats, equipments are streamlined.

Here's some overview:

• First 6 games were 2D games, the best of those bunches are Final Fantasy 6. Great story, great villain, great music
• original FF7 is the one popularized the JRPG genre, and probably broke the base between older 2D fans and newcomers. It has memorable characters, music, story about eco-terrorism. The gameplay revolves around materia system, it's like logic system where say if you connect Fire magic with All-effect and 2x-effect, you can casts double Fire magic that hits every enemies, etc. FF7 Crisis Core is one of the best FF spinoffs out there, while FF7 Remake is a 'remake'. It's advised that you finished the original FF7 before playing those two.
• FF8 also broke the base. The game is more romance-centric in some way, but still sci-fi. The gameplay revolves around junction / draw system, where you draw magic from enemies to junction it to your stats.
• FF9 is kinda back to original game. It's more high-fantasy setting, and was released during the end of PS1. It wasn't as popular as FF7 or FF8, but there are definitely fans. I had hard time getting into it, because the animation is kinda slow, but maybe I should replay the HD version
• FFX is very well received, it's a sci-fi romance story that takes place in south east Asian-like tropical islands. The first FF game on PS2. FFX has a sequel, FFX-2, which is also well received
• FF11 is MMO, I don't play MMO, so I have no idea about it.
• FF12 is great, it's more political than usual FF games, because it's written by Matsuno, who made Tactics Ogre and FF Tactics. The gameplay is bit weird, bit MMO like.
• FF13 was not well received, the only mainline FF game on PS3. It spawned two other games FFX-2 and FFX Lightning Returns. The main complaint about FF13 was that the story was incomprehensible, the game is very linear, and the battle mechanics is very confusing. I think what happened is that
  • they used tons of opaque in-game terms (Fal' Cie, La' Cie), that's barely explained until very late in the game.
  • the game also opens up very late, there's a one large wide region for you to roam around and engage in enemy encounters, but they only give it to you very late in the game
  • the combat wasn't explained clearly, the paradigm shift system is actually fun, and a step up from ATB
  • annoying characters, they focused too much on Hope and Snow. Hope is a whiny child, but he's a child, so it's ok. Snow on the other hand, is just an annoying character who likes to talk about himself.
• FF14 is another MMO, I don't touch MMO
• FF15 is kind of a mess, it was in development hell. I like the roadtrip story, where you just drive around. The open world is bit sparse and serves mostly for enemy encounters. One of the main issue is that some of the stories are gated behind DLCs. The gameplay is bit more weirder than normal ATBs. I like this game, but not as much as others.
• FF16 is great. Devil May Cry combat, very streamlined and nicely paced story, those huge spectacle Asura's Wrath-esque battles, etc. This game is my current GOTY.

There are other spinoff games, e.g. Final Fantasy Crystal Chronicles, World of Final Fantasy, but they are mostly spinoffs, mostly for fans who want more after playing the mainline.

But there's one that I want to recommend, and that's Final Fantasy Tactics. It's a strategy RPG and it's amazing. There's an updated version released on PSP, called Final Fantasy Tactics: The War of the Lions, which is probably the one you should play.

There are rumor swirling around about FF9 and FF Tactics remakes, but can't say anything until we see it.