Skyrim:Take Up Arms

Quick Walkthrough
# Locate the headquarters of the Companions.
# Find and talk to Kodlak Whitemane.
# Prove yourself worthy against Vilkas.
# Make yourself useful to the higher-ranked members.

Detailed Walkthrough
Find the Companions
On your way to Whiterun, you may come across members of the Companions, specifically Aela the Huntress, Ria, and Farkas, battling a giant among the sparse crops outside Pelagia Farm. If you want to participate in this fight, make sure to head straight to the farm once you arrive in the vicinity of Whiterun. If you delay for even a short time, the Companions will have already defeated the giant before you can join the fray, resulting in Aela taunting you for your lack of assistance. If you delay for too long, you may miss the Companions entirely, as they will slowly walk to the main gates of Whiterun and enter the city after the battle. Note that neither of these results has any negative consequences in your dealings with the Companions. If you are present when the giant dies, Aela will actively seek you out and will, depending on the outcome of the battle, either commend or taunt your battle skills. In both cases, however, she will tell you about the Companions and point you towards Kodlak Whitemane in Jorrvaskr.

This quest can also be started by speaking with select Whiterun citizens or Jorrvaskr residents, who will suggest you go speak with Kodlak.

Jorrvaskr
Jorrvaskr is a legendary mead hall and the oldest building in Whiterun; it is located in the Wind District, up the eastern set of stairs from the Gildergreen. Inside the building, you will witness a friendly brawl between newcomers Athis and Njada Stonearm while the other members cheer. The harbinger of the Companions, Kodlak Whitemane, will most likely be in the northern room of the Living Quarters, busy talking to Vilkas about what seem to be important matters. When asked about membership, Kodlak will consider you worthy, while Vilkas will disagree. After a brief argument between the two, Kodlak will ask about your battle skills. Regardless of your answer, Kodlak will order Vilkas to escort you to the yard for a quick sparring session. Follow Vilkas to the yard and pick up the fight. Make sure not to use magic, a bound weapon, or an enchanted weapon, for if you do Vilkas will temporarily stop the session. If you only have enchanted weapons, unequip them and use your fists. Take a few swings at Vilkas and prove your worth. Once Vilkas asks you to stop, accept your status as a 'whelp' and the fact that you will run errands for now.

A Few Errands
Vilkas will hand you his sword for sharpening. Take it to the best smith in town, Eorlund Gray-Mane, who works at the Skyforge just up the stairs from the Jorrvaskr entrance. Eorlund will briefly explain how newcomers tend to get bossed around upon joining the Companions, and will then ask for a favor. He will give you a shield for Aela, who can be found talking to Skjor in the living quarters. Similar to the conversation between Kodlak and Vilkas, it becomes apparent that the Companions hide a few secrets, although none of them will share them with you at this time. When you hand Aela the shield, she will thank you for the favor and summon Farkas to escort you to your new quarters, where a free bed will be available. Farkas will introduce you to a few of the other newcomers and will also offer you a job, which will be one of his radiant quests. You can choose to ignore his request, as seven different radiant quests will now be available from the guild members, and you must complete at least one of them before your next major assignment, Proving Honor.

Achievements
One achievement is unlocked when you complete this quest:
 

Notes
In the living quarters, Aela and Farkas' dialogue will differ. If you met them earlier near Pelagia Farm, they will both recognize you; if you went straight to Jorrvaskr, they will just greet you like a newcomer.
At high levels Vilkas is quite powerful. Make sure to outfit yourself properly.
If you manage to drop Vilkas' life to zero during the test he'll qualify your performance as "most impressive" rather than just "not bad".
Athis and Njada Stonearm's brawl can potentially go on forever if the player first enters Jorrvaskr at a high level. Going into the basement will end the fight.

Quest Stages