Ok, as the resident FF geek, here goes:
I: Do not under any circumstances play the NES version of this game. On top of the terribly cumbersome interface (have fun buying 99 potions one at a fucking time), half the items and spells in the game are simply broken. Unless you REALLY want to see the series's roots, get one of the Remakes.
Your two options are either PSOne for accuracy - it keeps the original's AD&D 2nd edition rules of spell casting (fun fact, FF1 is basically a D&D game in all but name) where you get a set number of each level of spells, most of the crap in the game actually works right, and you get some decent cutscenes and general UI improvements.. or your other option is Dawn of Souls, the GBA or PSP remakes. They "modernize" the game a bit, it has a retranslation (much better!) and retcon a few little series nods, it gets a Cid, and a Chocobo reference, and it brings in the typical magic point based system of spellcasting. The downside to this is the game isn't really reblanced to address for the fact you have WAY more magical firepower and medical power than you're "supposed" to, and with a decent BM/WM wombocombo you can pretty much steamroll everything in the main game, up to, and including, the Warmech (the series's first optional uberboss). If you go this route, get the PSP version, it's nicer looking, runs smoother, and has an extra bonus dungeon on top of the GBA's bonus dungeon. Do not give Squeenix money for the shitty mobile port of this unless you have no alternative.
II: Cool idea, struggle in execution. The game's core mechanic is that it takes away leveling and introduces a brilliant mechanic where you get better by performing actions. The execution is fundamentally broken however. Get stronger attacks, hit things with swords, get better magic, cast that spell a bunch.. how do you get more HP? You get hurt. So you spend chunks of the game in a low level random battle smacking your guys in the face with your own weapons. It's a shame because the story's actually pretty cool for the era, it has awesome music (the Rebel Army Theme is amazing), and a clever conversation keyword mechanic. If you grit your teeth to play it, it HAS to be the PSP version, where they give you "level ups" and boost all your stats across the board every so often, which reduces, though doesn't completely eliminate, grinding (you're still gonna spend 5973458 hours honing Holy and getting a late game party replacement up to par before you go kill Emperor Mateus).
III: The first truly good FF. Great game. Get the DS/PSP version, it's a nice modernization and it replaces the generic template Onion Knights with an actual cast of PCs (this is the cannon version now as well). Just make DAMN sure you're ready for the final dungeon, it's a multihour sink filled with about 15 bosses, and the End Boss is a MASSIVE powerspike - unlike every other FF ever, the last boss is legitimately far and away the hardest thing in the game (save the optional bonus dungeon uberboss). Oh, and there's no save points anywhere in there.
IV: One of my favorites in the series. As long as you avoid the original US release (as FFII), which they gutted for simplification and ease, you'll be fine. I personally prefer the PSP 2d remake but the DS 3d one is solid as well, but not as true to the source material as it changes a lot of the "rules" of some of the encounters. Has a better translation though.
V: Origin of Gilgamesh. That's all you need to know about this game.
(kidding aside, the game is awesome, and it's the job system perfected)
Fuck the mobile remake and it's terrible shitty artwork with a Masamune. PS1 port of this game is hot garbage as it's got input lag. Get a fanslation of the SNES game, or the GBA version.
VI: Is a masterpiece and everyone should play it. Ditto for V with the mobile port (even harder), and PSOne (the input lag here renders Sabin's Blitzes all but unusable). The GBA version is the definitive version with one major complaint, the music is nowhere near as good as the SNES version, and the Opera Scene really deserves Aria di Mezzo Carattere done right (plus the game's soundtrack is amazing anyways). But the rest of the GBA version is better, sooo.
VII: Graphics are unbearable, translation is a dumpster fire, by far the easiest game in the series, yet this game is still somehow amazing in spite of all that. If Squeenix would ever give a hint of a date at the remake i'd suggest waiting, but as is, Steam version, plus graphics and text mods is probably the best way to go if you're willing to put in the legwork to get it cobbled together.
VIII: Skip it and play IX. Failing that, STEAM STEAM STEAM since you can enable EZ mode to avoid the horrific work of drawing spells after every fucking boss fight once you realize how stupid and broken the game's battle mechanic is (tip, don't kill things, they level up with you and get stronger than you do respectfully). Only redeeming factors of this game are Zell's a Dudebro, Selphie's name has become retroactively hilarious, Triple Triad is one of the best minigames ever, and the soundtrack is PHENOMENAL and deserves to be in a better game. I really should replay it someday, honestly, to give it another chance, but ugh.
IX: Wonderful, underated gem of a game, and IMO the last TRUE Final Fantasy game (some of the later games were great, but this marks the point the series ends its classic builds, for better or worse). The last couple hours of the game is an out of left field what the fuck, but the ride getting there is a blast.
X: Yep, HD remake. Great game. You'll like X-2. It's fanservicey as hell and it's a love it or hate it game. You'll love it. It's completely ridiculous and fun.
XI: hard skip unless you want to stab rabbits for 2 weeks and every version of Shantotto and her stupid laugh should be burned with a prejudice. Not even sure it's still online to be honest.
XII: Zodiac Age is the definitive version of this game. I need to buy that to replay it. I only rank XII lower because it's not what it was MEANT to be (massive executive meddling). But for what it is, it's a good game.
Tactics: Added to the list because it's one of the best games of all time, and since you like Fire Emblem, you'll like it. Just make sure you're ready for Riovanes Castle.
I'll end there since I know you've played XIII.
I really need to give XIV a try, I've heard good things about it since the hard reboot.