Some questions

[JukeboxHero]

Levequests are a viable leveling method for solo play, and are very different from the actual quests within the game. The quests can be found pretty much all over the three towns, whenever a character has an exclamation point over their head. You only see the exclamation point if you are of a level to do the quest though, so starting out, you won't see many at first. Also, you can't see quest givers on your map, as I recall, which can make it a pain to actually find some of them.

The main difference in quests in FFXIV from a standard MMO is their presentation and purpose. As far as presentation goes, they tend to actually have story and character, which I personally find nice, though some may find it to be tedious I suppose. There is some nice cutscenes/lore/story in some of the random quests though. What I mean by purpose is that unlike most western MMO's you do not level from quests. Most of the XP from quests is pretty pathetic outside of the main story ones. You do get some nice gear for lower levels from some though.

In short, Levequests for leveling, quests for story progression and breaking up the pace of the game a bit.

As far as leveling multiple classes, it is necessary to an extent, especially if you are playing mainly solo. Most of the really useful stuff you can pick up at pretty low levels though (Cure for instance). I've heard stories of the more established players being kind of mean in parties when it comes to players not having specific things leveled and what not, but I really can't comment on that.

Aaaaaand cue someone else now to give a much better explanation to your other questions...