TAKE UP ARMS
The Take Up Arms quest is obtained from Kodlak Whitemane in Jorrvaskr.
Train with Vilkas
Give Vilkas's sword to Eorlund
Bring Aela her shield
Follow Farkas to your quarters
While you've been surviving by yourself just fine up to this point, you'll have to prove yourself to Kodlak and The Companions before they accept you into their ranks. Follow Vilkas out to the training grounds to test your mettle against this trained swordsman. Be sure to equip a weapon and swing at Vilkas, who gives you Vilkas's sword to be sharpened with Eorlund at the top of the steps nearby.
Show Vilkas' sword to Eorlund to learn more about the group's leadership (or lack thereof). As you turn to go, Eorlund asks a favor of you: to bring Aela her shield. You can find Aela inside Jorrvaskr's living quarters. After the talk with Aela, Farkas will show you to your appropriate sleeping arrangements and also give you the quest Hired Muscle.
Speak to Skjor
Speak to Farkas
Retrieve the fragment
Return to Jorrvaskr
After returning to Farkas to complete Hired Muscle, Farkas tells you that Skjor is looking for you. Find Skjor in the main common area of Jorrvaskr to find out what he needs. Skjor has deemed the task of finding the great weapon Wuuthrad worthy enough to be your Trial. With Farkas designated as your Shield-Sibling (which hereby replaces any Follower you may already have in tow), you must speak to Farkas again and pick his brain on a number of subjects. Let him know you will meet him at Dustman's Cairn to give you some preparation time for this journey.
Dustman's Cairn teems with the undead, though the brawny Farkas can certainly lend more than just an extra body -- he is rather adept at killing the draugrs himself. Read the tome entitled The Battle of Sancre Tor on the table for an increase in Two-handed weapon skill.
Deeper in, you reach a room bigger than the rest - one with an Arcane Enchanter and seemingly no exit. Head into a niche filled with potions and pull the lever. This closes the gate on you and traps you inside. Farkas unleashes his secret power to defend himself against the ambush. After being freed, proceed along the dusty corridors to the Dustman's Crypt. On the way, you will have plenty more Silver Hand humans to eliminate.
Once in the crypt, though, most of the enemies you encounter are of the undead type. Push through the sporadic undead appearances until you're stopped by a door that absolutely requires a key. Search the chest in the same room for Dustman's Cairn Key.
A change of enemies: Skeevers start to make frequent appearances and you'll even have to squash some Frostbite Spiders. Put down a few more draugrs and head into a room that differs from the rest of the ones you've visited thus far, in that this one features the Fragment of Wuuthrad you've been seeking. As well, head to the back for another World Wall to learn another Shout: Fire, Fire Breath.
The coffins around the room begin to stir and out comes draugrs, one after another. Put down these draugrs for a while for extra xp and then leave via the side passage, which skips the arduous task of backtracking and deposits you right by the exit.
Fast travel (if you want) back to Whiterun and approach the steps of Jorrvaskr on which Vilkas already stands. Vilkas takes you around the back to complete the Companion induction ceremony. Before you part ways with Kodlak, he tells you to visit Eorlund for a Skyforge Steel weapon of your choice.
At this point, a great number of Companion-related quests open up. You can now speak to other Companion leaders for more work, but you can only have one active at a time.
THE SILVER HAND
Talk to Skjor
Meet with Skjor at night
Enter the Underforge
Participate in the blood ritual
Talk to Aela the Huntress
Kill the Werewolf hunters
Talk to Aela the Huntress
This quest allows you to become a werewolf). Becoming a Werewolf, increases your health and is reversible if you care to dabble in the transformation.
Before you start this quest you should probably part ways with your follower as you will gain a new one for the quest.
After becoming a Companion, you can speak to Skjor or be referred to him by other leaders after you have completed other quests. When you find him, he would like to speak with you in private at the Underforge. Head to the Underforge, making sure it is night time (use the wait feature if you have to).
There is a pretty clear outline along the rock wall that indicates where the secret entrance is. Head inside the Underforge, and when you are ready tell Skjor to begin the ritual. When you come to, you will have already reverted to your human form, although you can transform into a werewolf by selecting Beast Form under Powers in the Magic menu. Keep in mind that you’ll hit a cool down period.
Aela mentions a Silver Hand camp nearby that must be exterminated. Head into the stone bunker at Gallows Rock. (If you do not go straight to Gallows Rock, Aela will follow you wherever you do go, though she doesn't seem to help with enemies that aren't related to the quest). The Silver Hand have odd tastes in decor, with werewolf heads ceremoniously placed on pikes and put on prominent display. Never mind that warning because the guys are wimps. Pull the chain to lower the gate and make your way to the leader, Krev the Skinner.
In the fight, Krev is accompanied by two archers; take them out first and focus your attacks on Krev to dispatch him. Another way to complete this objective (provivded that the beast form power is useable) is to turn into a werewolf before opening the doors to the chamber that Krev is located in. I would suggest to first mauling the silver hand working the tanning rack (the one seen above). Then to take out the archers and then krev himself. If one starts to get a bit overwhelmed/ low on health, use the Sound of Blood howl. This will make the Silver Hand run off in fear, allowing for the chance to feed upon bodies to heal and to extend bloodlust. Know that the howl does not last indefinitely and will have to be used again to renew its effects.
Speak to Aela afterward to finish the quest.
Skjor went in alone and was killed by the leader of The Silver Hand. Search his body for some good heavy armour and a great one handed sword for players around level 10.
Collect a Glenmoril Witch's head
(Optional) Wipe out the Glenmoril Witches
Return to Kodlak
After finishing Aela's quests, Kodlak will want you to collect a Glenmoril Witch's head.
The map marker will point you to a cave near Bilegulch Mine. When entering the cave, you'll find one Witch almost immediately in the central chamber.
After killing her and taking her head, you have the option of staying and slaying the other Witches or leaving immediately. (The extra heads are used to cure Lycanthropy, both yours and the Companions, if you decide to give up being a Werewolf.)
Either way, whether you fulfill the optional objective or not, when you return to Jorvaskor, you will find out the Silver Hand attacked in your absence, and there were losses on both sides.
PURITY OF THE REVENGE
Retrieve the fragments of Wuuthrad
(Optional) Wipe out the Silver Hand
Return to Jorrvaskr
For the next undertaking in Driftshade Refuge, Vilkas will be your shield-sibling, helping you hack down opposition as you trek through this enemy-filled dungeon.
Make your way into the Driftshade Cellar, making sure to avoid any traps. Do not free any werewolves that you come across (they will try and kill you) and slaughter the Silver Hand to complete the optional objective.
Eventually, you come upon an area where you can retrieve the Wuuthrad fragments. Return to Whiterun afterward.
GLORY OF THE DEAD
Attend Kodlak's funeral
Retrieve Kodlak's fragment
Give the final fragment to Eorlund
Meet the Circle
Go to Ysgramor's Tomb with the Circle
Return Wuuthrad to Ysgramor
Get to the burial chamber
Speak to Kodlak
Put a witch's head into the fire
Defeat the wolf spirit
Speak to Kodlak
Attend the funeral for Kodlak, where The Companions have gathered and are paying their respects. After the funeral talk to Eorland who will ask if you have the fragments of Wuuthrad. After you give him the fragments, grab the final one from Kodlak's room, which is at the end of the hall where you first meet Kodlak. His room is located beside the chair he sat in when you talked to him about joining his faction.
Find the fragment in a bedside table along with his journal and a book on Sovngarde. After returning the fragment to Eorland, he thanks you and tells you that the remaining members of the Circle are in the Underforge.
When you enter the Underforge, they are discussing a way for Kodlak to enter Sovngarde. Afterwards, you receive Wuuthrad from Eorland.
Make the long trek to the burial site of Ysgramor and give the axe to the statue to open the passage to the tomb. Proceed inside and cross swords (or staves, bows, magic, or whatever) with a variety of baddies, including Companion ghosts and some Frostbite Spiders. You'll eventually enter the burial chamber and find Kodlak's ghost, who will ask you to put in one of the witches heads into the fire to release Kodlak's wolf spirit.
You have to fight this beast. After defeating the wolf, talk to Kodlak to be given the title Harbinger.