The Companions

On the way to Whiterun you may find some warriors fighting a giant. Either if you help them or just watch them they will offer to take up as a Companion. In order to become one you have to visit their camp inside Whiterun at Jorrvaskr. Speak with Kodlak Whitemane, the leader of the Companions and ask if you could join them. A small test is required to see if you are eligible to be one of them.
  Even if you don't come across the giant-fighting event still you can just visit Jorrvaskr and speak to Kodlak.

Take Up Arms

  So you are about to fight Vilkas to prove you are talented enough. Go outside to the backyard and let the fight begin. You cannot use any spells during the battle otherwise you will fail it. Upon defeating Vilkas, he will sends you to sharpen his blade with Eorlund, the loyal smith of the Companions. He is just up on a higher platform next to Jorrvaskr. He takes the sword and advises you not to be a servant of anyone, however afterwards he do asks you a favor, to take Aela's shield back to her. Follow Farkas who is going to show you your room.

Trouble in Skyrim

  Your first job is straight from Farkas, who tells you to kill the leader of a bandit group located in the Halted Stream Camp. The leader resides in a mining cave inside the camp. Lots of iron ore veins can be mined on your way inside. Several bandits surrounds the room with the leader in it, so get ready before entering the fight. Upon killing the leader don't leave just yet! Look around, there's a spell tome of Transmutation on his table.
  This is a perfect place to gather an insane amount of gold ore by mining all the iron and casting the Transmute spell to convert all of them into silver and then into gold.
  Return to Farkas and take your reward.

Proving Honor

  Farkas mentioned that Skjor had been looking for you so seek him out and ask what did he want. The location of a piece of the Wuuthrad had been revealed and that's what you have to investigate along with Farkas. Farkas replaces your current follower if there's one.
  Go to the Dustman's Cairn where Farkas has already been waiting for you and enter it. Draugr is what you will find inside, quite a lot of them. However get into a trap which was made by the Silver Hand. Farkas overwhelmed by them seems to die, but he transforms into a beast and manages the situation easily. He tells you that some of the Companions have a special gift and that you can get aswell if you are worthy for it.