There are many quests in the game that are given to you by the villagers or by finding something in the depths below.. I will try to list them in order of appearance. Keep in mind that because the Diablo engine is a random game creator, you will probably NOT see all these quests in one game. That does not mean you have done anything wrong, it simply means that that quest was not part of that game as it was created. Do not take this to mean that you can go around quickly and not look for things. As you can see, most quests are initiated by finding someone, or something. If you do not read the right book, or talk to the right people, it is possible that you will miss a quest that is available to you.
If you talk to the dying townsman near the church he will tell you of a terrible creature name The Butcher. The Butcher can be found on the second level. He is much tougher in multiplayer than he is in single, as are most of the bosses that appear in both. He can be found in a room that is easily spotted. It is square with one door and with lots of dead bodies littering the floor and stuck on pikes. He is very fast and usually requires a higher level character to beat him than the character you will have the first time down to level 2. In single player when you beat him he gives you a cleaver, which is a unique item, but in multiplayer the prize is random as it is with all multiplayer victories.
Talking to Pepin will sometimes give you a quest to solve the problem of the town's water supply. Search level 2 for a door which leads to a cavern. Once all the creatures are dead, return to Pepin and he will give you something.
Cain tells you the story of King Leoric. The King avoided capture by Diablo but ended up as a big nasty skeleton. In single player you run upon a door that leads to King Leoric's Tomb on level 3. On multiplayer it is much more fun since he is just waiting for you somewhere on level 3. He is always surrounded by a huge number of skeletons. Having the Holy Bolt spell is the easiest way to beat him. I was able to beat him once with a level 28 fighter in multiplayer. I think he has a sword that steals your life when he hits you because I have killed him with 4 Level 2 Holy Bolts before but if he can hit me it takes many, many more. Keep away from him and do not let him hit you or you will never kill him. In single player he gives you the Undead Crown, another unique item. In multi it is a random prize.
Talk to the innkeeper and he may tell you about his lost sign. Found on level 4, the last level of the dungeon, you will run into a little Carver named Snotspill. He tells you about "big uglies" that have something he needs. He promises if you return it to him he will let you live. Somewhere on this level you will find a group of overlords guarding a chest. Once you open it the tavern sign pops out. You can give the sign to Snotspill, after which you are attacked by a huge number of Carvers or you can return it to its rightful owner, who then gives you something. When you return to the area Snotspill and his Carvers attack you. I think the better choice is to take the sign to the innkeeper.
This is found on level 4-6, and it varies each time you get it although it is usually near the top. You read a book on another level that talks about the Chamber and then you find a staircase in the middle of some columns that lead to this section. You will encounter MANY skeletons (hence the name) and at times some creatures from lower levels like horned demons. This complex has a few rooms. You will eventually find a book of Guardian that you can read to get the spell, even if your magic requirement is not high enough.
Cain tells you a story of a brave warrior. Down in the catacombs you will find a small altar. In two other rooms you will find bloodstones which you need to place on this altar. After you do this a wall opens and you will need to kill a few more beasties. Your prize will be a piece of armor known as Arkaine's Valor.
You see a few rooms with a bunch of illusion weavers inside, but you cannot get into the rooms. You will find a book that tells of the Halls of the Blind. At this point you can get into the rooms through two doors. Once you dispatch the baddies, you will receive the Optic Amulet.
The blacksmith will tell you of a magic stone he knows of. You find it in the catacombs. It is sometimes hard to see because it is black and blends in in the darkness. Once you bring it back to him he will fashion something for you and give it to you as a prize.
The blacksmith talks of an anvil that can make a powerful weapon. I found this in the caves on level 10. Once you return the anvil to Griswald, he makes a sword called Griswald's Edge.
In the caves you can find a book which is unreadable. If you give it to Adria, she will explain the contents to you and ask you to get the mushrooms. Go back to the caves and get them. Go to Adria with the mushrooms and she will tell you of something that Pepin needs. Once you get this for him he will give you something.
Cain will tell you about the Archbishop who ran the church and turned to Diablo. You will find this happy man on level 15. He is surrounded by mages and some Hell Spawn. Dispatch them anyway you can and then take on Lazarus. I have used firewall on the mages, but use anything you want. The easiest way to kill Lazarus is to Stone Curse him and then kill him. You can also fry him with Apocalypse. The only other way is to capture him in a corner and beat him to death. (Those are the only three ways I've found) Once you do this you can descend to the level of Diablo.
Diablo can be found on level 16 and you can only get here by killing Lazarus above on level 15. Diablo's chamber is full of beasties. Kill them all and look for a lever. Once you find it, pull it and will open a wall that releases some more. Kill all of these and pull the next lever. Kill all the new monsters. You will then find two levers. Pull both and you have opened up Diablo's bedroom. Get all the magic you can and come back down. Walk up to the room and center yourself, but do not walk so far that you see the walls or Diablo will come say hi right away. Shoot all the magic you have into the chamber. After a few shots a pissed of Diablo will come out. Keep attacking him since everything that comes with him will die when he does. After he is dead, celebrate.