This quest is given by Farengar at Dragonsreach in Whiterun. You can complete it early by going to Bleak Falls Barrow before visiting Whiterun. Clear the dungeon and get the Dragonstone. Give it to Farengar as he is asking you to complete this quest to get a new dialogue option. However doing this will cause you to miss the conversation he has with his "reliable source".
After the Helgen prologue, you will find Guardian Stones on your way to Riverwood. You will be asked by your companion to choose one.
The senior members of The Companions are part of The Circle. To become a member of the Circle, progress through the Companions quest line, starting in Jorrvaskr/Whiterun.
Both are earned by progressing through the Dark Brotherhood quest line, which starts by joining the Dark Brotherhood after visiting Aventus Arentino in Windhelm and following his orders.
Obtain any Daedric artifact for "Daedric Influence" or fifteen of the sixteen total Daedric artifacts for "Oblivion Walker". The Skeleton Key does not count. Complete the indicated quests to receive the corresponding artifact.
Speak with the innkeeper in Riverwood, Orgnar about receiving training in magic and he will direct you to the college. Travel north toward Whiterun and speak to the carriage driver at the Whiterun Stables. Select "Winterhold" (for a nominal cost). Otherwise, speak to any innkeeper or spell merchant for the same dialogue option in other towns or cities. You can use any Stable to reach Winterhold. After arriving in Winterhold, travel north toward the College. When you get there, you will be stopped by an elven mage called Faralda and receive the "First Lessons" quest. She will direct you to cast a specific spell (which depends on how you responded to her initial inquiry). If you do not know the required spell, she will offer to sell it to you; however the cost may be more than Farengar at Dragonsreach in Whitehold). Once you cast the requested spell, she will speak for a short period of time and earn "Gatekeeper". Note: There are two alternate ways for you to get in without casting a spell for Faralda. Either persuade her to let you in (requires 100 Speech skill, 70 Speech skill with the Persuasion Perk (a 50-Speech perk), or the Amulet of Articulation from the Thieves' Guild quest line); or completing the required Dragonborn quests which opens a new dialogue option where Faralda asks you to perform a shout.
This can be earned after you complete the Helgen prologue. In Riverwood, take one cabbage from the cart in front of the trader in Riverwood and either purchase a Salt Pile from the innkeeper or find one in the barrels around town. Enter the house you were invited to after Helgen (Alvor and Sigrid, or Hod and Gerdur). Use the Cooking Pot to create Cabbage Soup. Fast travel to the Guardian Stones, then move east to Embershard Mine. Kill the single bandit outside, then find a Woodcutter's Axe across from the Wood Chopping Block (near the rock wall). Select the block to get credit for this step. Finally, enter the mine. Remain on the right due to traps along the left side heading down the slope. You will eventually enter a small cavern. Kill the two bandits to your right then go to the woodpile by the fire to find a Pickaxe. Take it and mine the Iron Ore Vein located directly in front of you.
Go to Riften at the lower right corner of the map. If necessary, you can travel there by spending about 50 gold for a carriage at any of the city stables. Once there, go to Temple of Mara and talk to Maramal. Ask him how marriage works in Skyrim, then purchase an Amulet of Mara from him for 200 gold. Equip the amulet, then find someone interested in marrying you. Good places to search are Riverwood or Whiterun. Return to Maramal and tell him you are ready to marry. You will be married the next day.
You must build three wings onto each of the three houses. Simply building a house on each of the three plots of land is not enough. Note: You do not need to fully furnish each house to qualify. All that is required are the "shells" for a house, including the basement. To prepare, stock up on iron ore and ingots, as well as a large amount of gold. The other materials to build the wings can be acquired by having a steward for each homestead. The stewards can be ordered to purchase building materials, which then get deposited into the lockbox next to the drafting tables. Lumber must be cut yourself or purchased from lumber mills. Having the steward saves you the trouble of carrying around excess material, allowing you to focus on getting the iron ore and ingots to smith into building materials (locks, hinges, iron fittings, nails). Iron ore is the only building material that you will have to actively search for. You can purchase iron ore and ingots regularly from the Solitude Blacksmith, Whiterun Warmaiden, and Whiterun Skyforge's Blacksmith. You can also steal some every few days from inside the Solitude Blacksmith's shop, on one of the shelves.