Find Jorrvaskar [Left] and talk to someone inside, or have a random encounter with a group of Companions out in the wilderness [Right].
So you want to be a “Companion”? Well, there’s two ways of going about it.
It's very likely that as you first approached the town of Whiterun you encountered a group of people fighting a giant/troll/animal out in a field. Hell, chances are you tried to help! Once the beast is slain, talk with them and you will receive a **Miscellaneous Quest** called **"Speak with the leader of the Companions".**
Or you can visit their place of rest known as Jorrvaskr in **Whiterun** (it's to the right of **Dragonhelm** on your map) and speak to someone inside about joining. They'll tell you to speak to Kodlak Whitemane, who can be found downstairs in the living quarters.
This is the best **** option as you're not relying on a random encounter to trigger.
Once you find Kodlak tell him you’d like to join the Companions. He will look you over and say you may have what it takes. He’ll then ask how you are in battle. Answer however you want and after a brief interruption by Vilkas, he will tell Vilkas to go test you out. This ends the MISCELLANEOUS QUEST that brought you here and starts a PROPER quest: “ Take Up Arms “.