Here is a quick quest guide on how you can accomplish the Take Up Arms quest in Skyrim.
First, you need to talk to Kodlak Whitemane so that you can join the Companions. Kodlak will require you to tag along Vilkas so that you can assess your performance in a battle. Just go where Vilkas go. Vilkas will go upstairs and out the backdoor of Jorrvaskr so go along with him. When you reach the courtyard, talk to him. When you do this, be prepared to fight Vilkas using melee attacks only. You need to use melee because magic attacks usually causes a malfunction in this quest and you might end up being unable to pursue this quest line any further.
The battle will usually just take up a short period of time but it will be very extreme especially since you will only be using melee attacks. When you defeat Vilkas, he will consider you worthy to be a member of the Companions and he will assign you to take his sword to Skyforge where you need to deliver it to Eorlund Gray-Mane. Skyforge is just north from your position where you can see a giant bird statue. Just walk on the Northern side of Jossvaskr. When you come across a stone stairway, walk up and go right to Skyforge.
When you talk to Eorlund, he will give you a shield that you need to bring to Aela the Huntress. Take the shield and go back to the courtyard. Aela is commonly found in one of these three places – the courtyard, inside Jorrvaskr Hall, or down in the living quarters. Look for her and when you find her, talk to her. Give her the shield Eorlund asked you to deliver.
After talking to Aela, she will direct you to go to Farkas. Farkas will be the last one you need to talk to finish the exchange. When you find Farkas, he will tour you around Jossvaskr. He will show you where you can lie down and rest. Pick any empty bed and claim it as your own. This will be the end of the Take Up Arms quest.