I've been thinking lately about how a speedrun of this game might look, and though I don't actually intend to run this game (I prefer a more leisurely pace), I have thought of some ideas.

First of all, I would consider the speedrun to end when the main final boss is killed and the credits rolled. The post game doesn't seem well suited to speedrunning, especially if you include Ibag Tower. (20 floors with dangerous monsters and Diomente not working? I don't think so.) For a more complete playthrough, one could aim to get all the tablets, which requires exploring every dungeon.

The main route I'm thinking of is as follows:
1. Play through the early game, ignoring side quests. (Anyone wanting to run this game should have enough of the maps memorized to skip the Underground Goblin Map, and I'm pretty sure you don't otherwise need to go past the first floor of the Grotto or Tree.) The boss of the mine does have to be fought, however. (Remember, status ailments work on bosses in this game.)
2. When you reach the church basement, go straight to the deeper floors and level up off the Lesser Demons. At a minimum, learn Diomente.
3. Run through the rest of the game. If needed, you can use the Devilish fountain and then level up off Greater Demons.

Now, as for party selection:
Mages can learn Diomente at level 13. This allows you to teleport around easily. Also, Miracle can make a big difference in boss fights.
Clerics can learn Honey Restorer at level 7. This can save time needed to leave the dungeon, but remember that it needs to be unlearned after every casting.
A Fairy Alchemist might be a good idea. I don't normally recommend this because of slow leveling making High Purity Synthesis weaker, but that really isn't a concern from a speedrun perspective. As a Fairy, however, you get high Agility (to act first more often) along with Mahamaha (for running away quickly from time-wasting battles) and status ailments.
A Brawler might be a good attacker. A Fairy Brawler might actually be a good choice thanks to high Agility, Lower AC, and the fact that you can get a Star Rod if you take a detour into Hostrana. This gives you a nice attacker without being RNG dependant. On the other hand, a Dragonewt gets a breath attack and high Strength. (Note that the Brass Knuckle is ineffective against Gods (like the final boss), but a first level Cleric spell can counter that.)
Dragonewts are really good on reaching level 8, especially if you have more than one, and especially if you stick to fast-leveling classes.
Is there any need for someone with Thief skills? Unless there is a mandatory locked door that requires Thief Skills to open, I think the answer is no, especially with High Portal existing. (But your Alchemist doesn't really *need* to go past level 9, so the experience might be better distributed to other characters.)

Edit: One more thought. A class I never used, the Hunter, might work well. Getting the extra attack each round on a disabled enemy will make battles end faster.

There's also the factor of EX skills and which are useful from a speedrun perspective. Remember that what is good for a normal playthrough might not be good here and vice versa.