I've been a fan of the series since late 2004 and spent 2005-7 collecting the various ports to play through the whole main series, barring the MMOs. That said, I am familiar with FFIX, having played a month of it in 2010 and then watching all of the cutscenes. I plan to do the same with FFXIV in a few years. The following are my brief thoughts on each of the main games.
1. VI - First played on the GBA in 07. It was love at first site. The characters and moments stuck with me, and I adore that soundtrack. A definitive classic in my book.
2. VII - First played in 05. Like VI, stole my heart, despite how ugly it looked. Probably my favorite cast in the series. I have played the game to death; seen every date scene through my five playthroughs.
3. V - First played in 06 on the GBA. I love the job system and the humorous style. One of the best for gameplay.
4. II - First played in 04/05 (can't remember exactly) on the GBA. Broken? Yes. Fun? Hell yes! It's my real "first" FF--not first played but first played all the way through. It holds a special place in my heart.
5. IX - First played in 06. One of the most solid experiences with the only real flaws being the far-too-slow ATB and automatic Trance.
6. XII - First played in 06. As mentioned, the major design flaw was focusing on Vaan and not Ashe, the central protagonist of the plot. I love the setting, design, and dub. I just wish the plot was better paced.
7. VIII - First played in 06. I love the aesthetics, and I love the Junction System. ...Drawing, however, can bite it. The plot is a bit of a mess but has led to some awesome discussions with my best friends.
8. IV - First played in 05 on the GBA. It may be a classic but it is incredibly shallow. Unlike II, it doesn't have the guts to kill the cast, leading to a ridiculous plot that undermines everything it's trying. And the plot is important, because they skimmed gameplay for it. As inventive as the ATB was at the time, it doesn't help much if you can't do anything to customize the party.
9. I - First played in 05 on the GBA. Nothing to really dislike, but this oldie isn't bringing much to the table either. Love the ending, though. Best in the series.
10. X - Played for the first time in 04. First one I touched though I did not finish it until 2010! ...I always got bored and quit at the Calm Lands. I hate the main characters. Tidus annoys me and Yuna isn't too much better. The plot isn't bad, just insufferable with these two at the helm. I like the gameplay but not the Sphere Grid, though I've seen worse (i.e. XIII's Crystarium).
11. XI - Played in 10, on PC, for a month. I then used YouTube to watch ALL of the storylines. The plot is the series best, hands down. Go watch the cutscenes, now. Do it. Sadly, I couldn't quite stand the game itself. I had some fun at first, but the moment I left the starting areas, the monsters ramp'd up in difficulty to the point that you NEED a party to survive. The game is better for solo now a days but still not terribly likely.
12. XIII - Played in 10. I like the battle system, but sadly a battle system cannot carry a game for 50 hours. At first, I was hooked. It streamlined the experience to make a modern experience, which is not inherently bad. Sadly, they forgot the streamline the length. This game should have been 20-30 hours of intense action and drama with an awesome battle system. Instead, it drags on and becomes unbelievably painful to digest.
13. III - First played in 06. Beyond the soundtrack, I can't think of anything nice to say. The job system in the DS version punishes experimentation, and the "updates" were shallow by only giving names to the characters and touching up little else in a simple plot.
Of the spin-offs and sequels, I have only played FFX-2 and Tactics. I love and hate aspects of both. I adore Tactics's plot but hate its version of tactical gameplay, being more of a fan of the Fire Emblem style. Conversely, FFX-2's godsend is its amazing battle system, though the rest of it is embarrassing. Including these two games, my thoughts on the series:
-Game Play (battles and character building/customization systems)
-Setting (the world/towns/NPCs, art style, musical score)
-Story (plot and major storyline characters)