With a couple of exceptions, quests in Morrowind are almost always of the same type: go somewhere, do something, report back to the quest giver. Some are simple shopping expeditions, while others are a bit more complicated, but that is the general pattern.
Secondly, the developers at Bethesda tried to provide alternative solutions to most quests. If you're used to bashing your way through whatever you run into, you'll probably want to sit back, take a deep breath and think about what you're doing. There are a lot of NPCs that show up in other quests that can't help you out very much if they are dead. I can't count the number of posts on the Morrowind boards that all seem to say the same thing: "I killed so-and-so and now I can't complete this other quest. HELP!!!"
Questing Rule #1: stop carrying your brains in your scabbard. This is not Quake or Doom where the rule is "if it moves, kill it." This is Morrowind where you can get what you want about half of the time by simply getting the NPC to like you.
Questing Rule #2: you have the ability to save games. Use it. I generally use four or five different saved game slots: before I accept a quest; before I enter a dungeon; before I get into a slugfest; after I accomplish the quest objective; frequent Quicksaves and any other time that I think a saved game might come in handy (like before trying something sneaky that might go south on me). If you're strapped for space on your hard drive, you can probably get away with just reusing the first five. But if you're strapped for a couple of megs, you might want to think about cleaning up your hard drive.
There are two major reasons for my emphasis on saved games. The first is simply common sense. If your system hiccups while you're in the middle of something, you've got a way to get back to where you were fairly quickly. The second is that it's fairly easy to bite off more than you can chew. For the most part, quests are not tied to your level, but to your rank in the faction that gave you the quest. Since your faction rank has only a passing resemblance to your level at best, your quest giver might be sending you off on a suicide mission.
Unlike Daggerfall, I have not come across many quests that have time limits. There are a few "escort the pilgrim" side-quests that have time limits (the NPC will let you know how much time you have left every 12 hours). Otherwise, the dialogue might give a sense of urgency, but there is no clock ticking on most quests. So if you're hopelessly outmatched, back off, gain a few levels and take it on again. Also and again unlike Daggerfall, most creatures are not going to respawn. This is especially true of NPCs. Once they're dead, they're out of the picture permanently. So you can knock off a few, go out and rest up, go knock off a few more, rest up some more, etc. Saved games make this much easier.
By far, one of the more frequently asked questions on the Morrowind boards centers around finding the place where you are supposed to go. In some cases, the quest-giver is kind enough to mark it on your map for you. But in most cases, especially for dungeon crawling, you're given directions and sent on your way.
Most of the time your directions are going to be enough to get you where you need to be. Other times, they won't. Since I have limited bandwidth available and since I'm lazy enough to not duplicate someone else's work (especially when it is light-years ahead of whatever I could do), you'll find an online and offline interactive map of Morrowind at the Unofficial Elder Scrolls Pages. Just enter the name of the place you need to go, click the button and you're ready to rock 'n' roll.
This is another area where Morrowind is light years ahead of Daggerfall. For the most part, the items you need are going to be in the possession of NPCs. There are a couple that you'll find on shelves or in boxes, but the vast majority are in an NPC's inventory.
As a general rule, an NPC will equip the best items he/she has. Weapons, armor, rings, clothing, etc. are most likely going to require a fight so that you can take it off of the corpse. But you can always try to pickpocket the NPC (save first) and maybe you'll get lucky. On the downside, this can also mean that the NPC is going to use that item against you before you can take it away. Another very good reason to save first.
The NPC who gave you the quest, almost without exception, is going to be the person to whom you report in order to finish the quest.
For the most part, successful completion of a quest gets you a 5-point increase to your reputation within the faction that gave you the quest and a 10-point increase to the disposition of the quest-giver. There may or may not be a few other goodies tossed in to the deal. For example, you might get a 1-point increase to your overall reputation. I've tried to cover all of these in the quest walkthrus.