If you've played the first Diablo, you'll notice that things work differently in D2. In each act of Diablo II, 1 through 4 (or 5 with LoD), they are broken apart into subquests. The default key is Q to bring up your quest log. At the beginning of each act, you'll be in a safe location with merchants and other NPC's. Speak to all of them before you begin fighting monsters to see if one of them issues you a quest.

In general, it's a good idea to speak to all of them again after the status of one of your previous quests has changed. They will most likely have something else to talk about, or issue another quest. Whenever any part of your quest has changed, or you enter an area needed to complete a quest, your quest log button on the bottom right turns red. Bring up your quest log and the quest will tell you what to do next.

Every quest gives a reward of some type, whether it's an item, an enchantment or the opening to another area of the game. Usually these will be given by the quest originator, but not always. After you complete a quest, its icon at the top of the log will turn to solid grey so that you know it's done.

In a multiplayer game, the person who made the game determines which quests are present for all other players who enter the game. Sometimes when you enter a game needing a later quest in the Act, you can borrow a waypoint from someone who casts a town portal to gain access to an area you don't have the waypoint to yet, effectively skipping a portion of the game. For your first time, it's recommended you experience the whole act, so that you know better what to do in Nightmare and Hell, where it's much more difficult.

Also when in multiplayer, if you're in a party and in the same Act as the other members, and they complete a quest that you still need, you also receive the credit and rewards. The requirement is that you're all in the same act. As a helpful tool, Blizzard also added a notification when other people not in your party or in another act have completed a quest. If the quest was needed by you, it's no longer available in the current game, so you'll need to begin another to finish it.

That's about it. It's an addictive game, especially when playing on Battle.net and being able to compare your items with others'. Here's to D3.