This is not the place to go if you're looking for help with a specific quest. This is the stuff you need to know and the stuff you should have access to before you even think about starting a quest.
Quests come in several flavors. There is, of course, the Main Quest where you find the missing heir to the throne and help him become Emperor of Tamriel. All of that will eventually be covered in the Main Quest Walkthrough section. There are also faction quests. The five factions (I'm not sure that the Arena counts as a genuine faction, but we'll pretend that it does) have a fairly long series of quests that all involve the same group of people and more or less fit into a story arc. Those will eventually be covered in the Faction Quest Walkthroughs. Thirdly, there are a boatload of Miscellaneous quests. These are ones that you might hear about through rumors, pick up through overhearing conversations, get by talking to NPCs or simply stumble into by entering a place or picking up an object while you're off exploring somewhere. I'm still arguing with myself over whether to do complete walkthroughs of these, but it's kind of hard to cover everything because these are where the modders enter the picture with the little one-off "go there and get that" kinds of things that fans like to put together. I'm kind of taking a wait-and-see approach, but will undoubtedly cover some of the more interesting ones that I run into. There are also 15 Daedric quests, one for each of the Daedra gods except Mehrunes Dagon (you're not on good terms with him in this game), that will get you a Daedric artifact. You're going to have to do one of them as part of the Main Quest (you don't get to keep the artifact), but the other 14 are available if you want to go seek them out.
And then there are your quests. These are the ones where you decide that you want to do something (collect a copy of every book in the game, assemble complete sets of each kind of armor, explore every Ayleid ruin, or even just see what's on the other side of that hill) and then set out to accomplish just that.
Aside from your quests, you start with a quest-giver. This is an NPC of some sort who tells you what they want done, (hopefully) points you in the right direction and then turns you loose. For some of the Miscellaneous quests, finding the quest-giver is part of the quest. For example, at some point you're going to pick up a Nirnroot plant. As soon as you pick up the first one, you'll get a journal entry letting you know that there's probably someone out there who would be interested in this plant and that you should seek him out.
On that same thought, your Journal is your bestest friend in the whole wide world. It will tell you things that you would not normally pick up on (like someone being interested in Nirnroots), fill in information that did not come out in a quest-related conversation, and keep track of the all of the people you need to see and things you need to do. Learn to love it because it will answer more of your questions than I probably ever will.
Your quest-giver is going to want you to do something. It might be to go talk to another NPC, to go fetch an item from some nearby dungeon, or whatever. That's the actual quest. After you do whatever it is that you're supposed to do, report back to the quest-giver and collect your reward. Aside from the artifacts that you get on the Daedric quests, the rewards are kind of middlin' (OK, the 10K gold for collecting all of those Ayleid statues ain't chicken feed) and it's the stuff you'll pick up while completing the quest that makes it all worthwhile (hopefully).
Which brings me to the subject of cheating. PC-version players have a the console and will find it much easier (and consequently much more tempting) to cheat your way out of (or into) a situation. Don't, or at least be VERY cautious about it. You can well and truly hoze yourself without intending to by activating quests or advancing quest stages before you have completed the prerequisite steps. How would you like to start the Main Quest with Jauffre locked inside Cloud Ruler Temple (where you can't get to him without even more cheating) while Martin is still inside the Chapel at Kvatch and niether of them have the conversation topics that you need? It's possible to do that simply by picking up a book in an area that you can't get into without cheating.
But here's an exploit you can live with. Quest objects cannot be removed from your inventory while they are still quest objects (which is a bad thing in some cases, but that's a whole 'nother gripe session). So when you're sent out to fetch something useful (or even if you just stumble over it), there's no reason why you can't use it for a while. And because it's a quest object, you can't accidentally sell it or drop it. Of course it can always be damaged and break, but the quest-giver isn't going to check the item's quality - just that you have it.
While you are on a quest or even out wandering around, you'll see markers on your compass. These markers point you in the direction of the object of the current stage of the quest. If the marker is red, the object of your quest is in another area (you'll have to do an area transition). If the marker is green, you're in the same area area as the object. A green marker on the overland map means that your object is out in the open. Other compass markers point you toward places that are not necessarily quest related, but that you have discovered during your travels. See page 6 in the game manual for a complete list of all of these icons and what they mean. The farther away from you something is, the dimmer its icon will be until it diappears from your compass altogether.