Be forewarned, it's been a long time since I've played this game, so I might say something stupid or leave something important out. Feel free to hammer away with the questions!
Also, for anyone wanting to speedrun this, reading the algorithms guide on Gamefaqs is a must. In fact, I haven't done so myself in a long while, so hopefully I don't say anything wrong.
First and foremost, I believe that from a speed standpoint, leveling up is a waste of time. In many RPGs, it will help you win faster but I think the difference here is more than lost by the time it takes to do so. Leveling up is only needed near the end of the game. I'll say where I think levelling should happen.
In general, it's a waste of time to buy any equipment. Armor tends to do stuff like improve your defense by 1 in this game. I'll mention any time I feel it would help to buy something. Curative items, like in any Square game and most RPGs in general, are ridiculous, so you'll want to spend your money on that. Mostly Life potions, as we'll get to later.
Picking up chests for the most part is also wasteful, of course.
It's cool to disequip people right before they leave, but I can't recall if it's ever actually beneficial. I'll assume it's not, although I did it when I was playing quickly.
The Float trick and the equip a shield trick don't come into play here.
Keep in mind that this is from a legitimate speedrun perspective. And off we go...
I would first like to get this out of the way and say that while it's not worth the time to do just this, Kain should typically be placed in the back row if possible since it doesn't affect his Jump. If you have the need, however, place him in the front with 3 back people for even more aggregate damage reduction.
I typically get the Black Sword or whatever in the caves; it's right in the way. When you get to Octomamm, let Rydia get off all the Chocobos she can before she dies. It's not worth reviving her. (If she somehow survives long enough, start using the Rod.) However, you'll want to have Tellah Life1 her during the vacuous part at the end. This is to make Antlion quicker with more Chocobos.
During Antlion, Cecil should attack for speed. Edward is Operation Human Shield. I think I've actually had him die to the upcoming Water Hag.
Against Milon and friends, focus on Milon with Cecil and Tellah. One Twin should take care of all the zombies. Don't heal him either, luckily for Cecil and his dark sword he's not undead yet. Against Milon Z, don't forget to swap your ranks! I think healing him works here but it's probably not worth it since the others have Fire and someone needs to heal. For the Dark Knight, just sit there.
Sadly, Yang is really tough at low levels. The quickest and easiest way to win this fight is not to kill or even attack him. Have everyone focus on staying alive, including Cecil with his Cure2. After Yang attacks you the third time, you can hit him once with Cecil and he'll "come to his senses."
I don't remember Baigan too well but I think Twin is good versus his Wall. For Kainazzo, his Big Wave will wreck your team, so the Lits need to be timed so this never happens. If you can pull that off, he dies incredibly fast.
Now we get to do some stuff with the Airship. The only thing I can think of is to go to Rydia's house and get the gear, namely the Tiara. I think the rod might be worth it too for any extra magic damage? I mean, it's right there.
In the magnetic cave, don't bother with any of the unequipping stuff, of course. When Dark Elf goes "ME CHANGE!", use Weak on the Dark Dragon and bash his face.
When I went through the game, I didn't do this, but in retrospect I recommend getting weapons in the Tower of Zot for Cecil and probably Cid. I know when getting the sword for Cecil, you fight guys who die easily to Ice3, and if and only if the same is true for Cid I'd do the same. The Magus Sisters is the first place you need a bit of luck to go quickly, and lack of bad luck if you want to not die. You basically need Tellah to Whatever3 whoever has Wall and hope it hits the fat one. Twice would be nice as well.
If you proceed like this, Valvalis is finally where you hit the brick wall level wise. However, this is where I figured out the tanking strategy that will be very useful for the rest of the game. Kain is your savior for this fight, but not because of "Jump, Kain!" It's because the evil baddies deemed fit for Kain to receive uber-gear. What you do is you immediately put all this insane armor on Cecil. Optimally, you would put him in the back, but for a speedrun it's best to put Kain in the back for 4x beatdown. Anyway, the only person you heal is Cecil. Whenever someone else dies, they get a Life potion. Basically, it gets to the point where Cecil Covers for everyone if you keep him at enough health. Now you can beat Valvalis, because she never targets more than one character at a time. She doesn't use Storm this fight but if she did, we'd be well prepared. ;)
While fighting Calbrena, it is important not to let them combine or you will not be able to do enough damage and go into an iterative loop. If I remember correctly, you should kill 2 of the easy front ones, then 2 of the back ones, then injure the last back one, and then kill them both really quickly before they can combine. The end of this fight also allows you to cure Cecil to the max, because he'll need it.
Try to get Kain to Jump against Golbez, but in my experience he is simply not fast enough. When Rydia comes, use her to Life people, starting with Kain, who should Jump repeatedly. Cecil can attack if he has the opportunity but I seem to remember him having to Life people as well. When you win, it is important to use Warp immediately and get the Dark Crystal.
Dr. Lugae and the whole next tower is strangely a pushover even at low levels. I can't remember if there are any special tricks to beating him quickly, but I think if you just sit there and let Balnab explode that's better than trying in vain to kill him first. I forget if you have Bersk now, but Lugae sucks, so it would be a good idea here.
Before fighting Edge's parents, get the treasure chest to the left, because I remember it's got LOTS of money. While fighting, just sit there to speed it up. Don't worry about healing, Rubicant's got you covered. Now most people think Rubicant is hard, but the truth of the matter is, like Behemoths, he mostly counterattacks, and with AE moves, which kill the tanking strategy. Rosa heals, Kain Jumps, Edge and Rydia hit him with Flood and Ice2 respectively. Now here's the key: Cecil can't either attack or parry, for tanking purposes, so sometimes you just gotta cast Cure1 or something. Because he's not attacking, feel free to put Cecil in the back along with Rosa and Rydia.
Now is where the game finally gets a bit non-linear. Also, we can now buy our final equipment with all the money we have. First go to the Sealed Cave and then go to the moon and come right back, for Excalibur reasons. Now go through the Cave of Summoned Monsters and don't heal from the damage tiles. Get the rat tail, and buy 2 Aegis Shields and the Summoner's Robe. In the dwarf town, buy 1 Gauntlet, 2 Rings, 1 Tiara, 1 Diamond Armor, and 1 Diamond Helmet. I think that's about right, I just looked at some ZSTs to try to figure out what I did. Now make the Excalibur. Now make some more. Outside Baron is the best place for the dupe trick. Because your guys are at 1 from the tiles, non-Cecil people can kill themselves off and it'll go by blazingly quick. I made 20 and won with 7, so 15 sounds like a nice number. I don't think any need to be made for money. I can't remember if I used them or Edge's magic against the Elements, but I'm getting ahead of myself. I will say that Edge should now typically be in the back row for Darting, given his poor defense.
Get FuSoYa. Rosa will then pimp him and take all his gear. Now in the Giant is where you want to finally level, so I guess the fact that you couldn't beat the Elements at this stage is irrelevant. Go to the save point and walk around. You are looking for those alerter guys, who summon dragons. Harvest dragons really quickly by using FuSoYa's Weak. If the timing is impeccable then the dragons won't even get to do anything; otherwise, they hurt you and you need to heal. When FuSoYa is low on MP have him Psych from your teammates. Whenever you run out of MP, go ahead and win the fight. You should level like a madman. I stopped when my guys got up to the low 40s. This took 2-3 harvests.
Now you can bash the Elements with all your elemental damage and decent stats and such, although Rydia didn't even have all the 3s yet for me. Against the CPU, you'll want to win the aggro way of course. I typically have FuSoYa start off with Meteo, Bersk Cecil, have Rydia use Titan, and Edge Dart of course. Titan should activate after awhile and kill the two little globes. Meteo should go off soon after for the win.
Now we have our final party. I have Kain in the front so Edge and the girls can be in the back. In the Lunar Subterranean, the chests we want are the Dragoon stuff, the Protect Ring, and the White Robe, chests without monsters. Once we get all this we can go to the menu and setup. I have Rydia: Charm, null, Tiara, Sorcerer, Diamond; Kain: Aegis, Gungnir, Diamond, Dragoon, Dragoon; Rosa: Lunar, null, Tiara, Sorcerer, Protect; Cecil: Excalbur, Aegis, Dragoon, White (yes!), Samurai; Edge: Middle, Middle, Samurai, Samurai, Diamond (Ring). I have tested and I'm pretty sure that physical defense does not matter at this point.
This is where I confess that Rydia, although my favorite character in a typical game, is complete dead weight. Zeromus counters with Nuke, which is not worth the trouble. Big Bang almost always kills her, even from full HP. In fact, sometimes Edge can die from full as well, and occasionally Rosa, in which case you lose. Before you use the Crystal, I typically have Rosa buff up Edge with Fast, but I don't know whether or not it is worth it because Zeromus just deactivates buffs later. Anyway, a lot of Attack, Jump, Dart, Cure4 happens. It's mainly a war of attrition. When he gets into his Meteo stage, you are home free, in the grand Square tradition of ending the final battle on a low dramatic note, a la Kefka being much easier than the other stages in FF3.
Whew.
