Missions are assigned to you by the Fighters Guild in the form of so called 'contracts'. In order to get one, just talk to one of the branch leaders: Azzan (Anvil), Burz gro-Khash (Cheydinhal), or Modryn Oreyn (Chorrol). If the one you choose doesn't have any contracts for somebody at your rank, you will be told to go to one of the other branches.
Joining the Fighters Guild is not a problem, all you have to do is to pay all the fines that you might have with the city guard. You can sign in by talking to either Azzan, Burz gro-Khash or Vilena Donton (Chorrol). You will be advanced to a higher rank when you complete a set number of quests.
A Rat Problem
Contract from: Azzan
Azzan, Anvil branch leader, will tell you to investigate a rat problem in Arvena Thelas' house. When you go to the Dunmer woman, it turn out that the problem is actually a mysterious animal that is killing her beloved rodents in her basement. Go there and kill a Mountain Lion, then go back to Arvena. She'll suggest you to go to Pinarus Inventius, who might just know how to get rid of these lions permanently.
Enter Pinarus' house and ask him for help. He'll offer to go hunting with you. Follow him outside the city gates to find four Mountain Lions. Get rid of them and return to Arvena. The problem hasn't been solved, apparently, as one predator is killing the rats even as you speak. Go to the basement and kill it, then. Arvena will tell you that she suspects her neighbour Quill-Weave of doing something that lures the animals into her basement.
You won't find out anything from talking to Quill-Weave directly. Sneak behind Arvena's house and wait until nightfall. You'll see Argonian planting some meat by the cellar's entrance. Talk to her. Now you have a choice: tell Arvena the truth (some Gold and bonus to Speechcraft), or tell her that Quill-Weave was innocent (Gold from Arvena and bonus to Acrobatics from Quill-Weave).
The Unfortunate Shopkeeper
Contract from: Azzan
The Guild will order you to solve Norbert Lelles' problem, whose shop is being robbed by some gang on a regular basis. You'll find it in Anvil docks. Ask the proprietor about the robberies. He'll tell you to stay in the shop overnight. Do so, and about 1 A.M. three bandits appear. When you kill them, find Norbert in a nearby inn, The Flowing Bowl.
The Desolate Mine
Contract from: Burz gro-Khash
The Guild orders you to deliver some weapons to Desolate Mine, north-west of Cheydinhal. Near the mine entrance you'll encounter some goblins. Go inside and find three warriors awaiting their weaponry. Give them what they need and help them rid the mine of all the goblins. You can join them or just wait for them to deal with the problem - either way is good as long as all of them survive. If someone dies, you reward will be cut. When you get a journal entry saying that the mine is clear, go back to Cheydinhal.
Contract from: Modryn Oreyn
Gro-Khash tells you to contact Vilena Donton in Chorrol. She will inform you that the Guild Champion, Modryn Oreyn, will be assigning your contract from now on. Talk to him to get the first one. Your mission is to find Maglir, guild member who didn't complete his contract and vanished. You'll find him in West Weald Inn, Skingrad. He'll tell you that the reward from the guild is not worth the risk of completing this one. Darn n00b, he is. The contract will automatically be passed on to you. Head to Fallen Rock Cave.
So, here's a small dungeon filled with zombies and ghosts. You have to find a Journal that is marked on your map. When you get it, go back to Maglir and talk to him once again. When you return to Chorrol, you have to choose whether to tell Modryn the truth or cover Maglir and tell that he was the one completing the quests. Either way is good, but the second one will save you some traveling later on.