Go to Whiterun, to the building named Jorrvaskr. On your map it's due east from the center - it's the big building that looks like (and actually is) a longboat turned upside-down.
First, you'll have to talk to Kodlak. He's usually in the basement of Jorrvaskr. After this, Vilkas will take you outside of the building (in the training grounds of Jorrvaskr), and you'll have to smash into him with your sword (or other weapon) for a bit until he thinks you're worthy.
After this, Vilkas instructs you to give his sword to Eorlund. This is the blacksmith found near the Skyforge (just north of Jorrvaskr).
In his turn, Eorlund will ask you to give Aela her shield. Aela is found in the basement of Jorrvaskr. Aela will instruct Farkas to show you your living quarters. Follow Farkas, and you've completed the Take Up Arms quest, unlocking the achievement of the same name.
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