You’ll sorta have to pick based on your tastes, but here’s a quick rundown from what I know. I don’t know all of them equally well (I know the older ones better), but here goes anyway. IMPORTANT: Games are not connected story-wise unless otherwise noted.

• FF1 - The beginning of the series. Barebones and open-ended by modern standards. You can think of this like a rudimentary videogame implementation of a D&D campaign, where you choose a party based on character classes and then go do quests to save the world.
• FF2 - The series’s first attempt at telling a significant story. Gameplay got weird though. Not recommended.
• FF3 - Character classes are back, but now there are more, and you can switch between them. Also there’s still something of a story too. The most polished NES release.
• FF4 - First SNES FF game. They tried telling a story, but it’s more complex and has more twists and turns (maybe too many? lol), and the gameplay is much better than before. Characters have fixed character classes and well-defined, plot-integrated roles. Your party can change frequently as the story progresses! Also introduces some other innovations like Active Time Battle. Has a ton of different versions, even more so than the other retro games.
• FF5 - A more straightforward (but definitely still solid) plot, but with an even more expanded character class system that you can switch around. And now you can combine abilities from different classes! Definitely recommended if you want to mess with creative combinations of abilities.
• FF6 - Much more complex and involved plot. Characters have somewhat fixed classes but the magic system is now more flexibly accessible. The first time the series went more steampunk/sci-fi instead of only doing traditional fantasy. (Contrast the Dragon Quest series which has kept on doing the usual Medieval European High Fantasy thing since forever.) Definitely recommended.
• FF7 - First 3D installment, with full-motion video cutscenes and all. Lots of steampunk/sci-fi and other influences. Character ability systems are still a hybrid of class-like features and complex customizability. This was many people’s first FF game, though fans of older games like to make fun of it for being overrated. The remake apparently changes the story somewhat (and might even secretly be a sequel or something?).
• FF8 - Went more heavily into the storytelling department. I don’t know much about this but for some reason fans of older games tend to dislike it. (Maybe the gameplay isn’t as great?)
• FF9 - Seems like a “neoclassical” FF in that it went back to its more fantasy-based roots. But it’s still 3D and you still get those prettyboys and such.
• FF10 - I don’t know this one that well but it seems to be solidly regarded. Also the first game to get a direct sequel, “FFX-2”. First PS2 FF game. Minigames and complex skill systems galore, from what I’ve heard.
• FF11 - An MMORPG, surprisingly. Something of a weird turn for the franchise, but it has its fans and apparently it’s still active.
• FF12 - I think this is the first mainline (and non-MMO) FF game to have real-time action for battles.
• FF13 - A massive three-part thing that’s very story-intensive. Apparently some people love the lead character while others hate her.
• FF14 - Another MMORPG. Initially it was a flop. Then they scrapped the game and brought it back significantly better and it’s now quite popular.
• FF15 - Apparently some dudes go on a roadtrip or something. I remember when the reveal was notable because holy crap you’re driving a car in an FF game. Besides this, I dunno much.
• FF16 - I even more dunno lol. If I recall correctly it may have something to do with being something of a prequel to FF1 but I’m not sure if I’m remembering the right thing. EDIT: I’m wrong; see the reply for more details.

Other FF games:

• FF Legend (1, 2, and 3) - Actually from the SaGa series but rebranded to sell better. Choose your species, and then there’s turn-based battles. For Game Boy. May be a bit arcane by modern standards.
• FF Adventure - Actually the first Seiken Densetsu game (Secret of Mana is the second). Action RPG for Game Boy.
• FF Tactics - Strategy RPG on PS1 with a quite complex story (lots of political intrigue and more). Very well-regarded. Original version has a somewhat iffy translation (leading to some unintentionally hilarious lines); PSP remake “War of the Lions” cleans it up. Character class system is here too.
• FF Tactics Advance - Strategy RPG on GBA. Somewhat similar gameplay but the story is totally different and more lighthearted.
• FF Tactics A2 - Strategy RPG on DS. Again, a different story.
• FF: the Four Heroes of Light - I don’t know much but I’ve heard this is a spiritual predecessor to the Bravely Default series.
• Chocobo’s Mysterious Dungeon - A game in the Mystery Dungeon franchise.
• Dissidia - Apparently it’s an FF fighting game.
• World of FF - Apparently it’s like a Pokémon game?
• Kingdom Hearts - this franchise is FF-style characters (and sometimes actual FF characters?) crossed with Disney IPs. Somehow this is popular.
• Chrono Trigger - I’m including this since it once shared a disc release with an FF game. Very well-regarded first game in the unrelated Chrono franchise.

If I didn’t mention anything, I probably don’t know anything about it.

Also, sadly, none of them are available DRM-free (legally, anyway), but what can ya do.