A Friend in the Guard
Head to the barracks in the Viscount's keep to talk to Aveline. This will unlock her as a companion. She can be very useful as the party's tank, so it's recommended you go talk to her as soon as possible. Also accept her task to get receive the quest "The Way It Should Be".
A New Home
Head to Gamlen's home in Lowtown. Make sure to interact with the Writing Desk to read a letter for you, and talk to your mother and then to Carver (or Bethany) about the Birthright companion quest. Also collect tomes from one of the chests in the house to complete Lothering's Lament and The Far Cliffs of Kirkwall.
A Business Discussion
Talk to Varric in Lowtown's Hanged Man pub in Lowtown. Just enter the pub, and interact with Varric to complete this quest. This unlocks the "Rumors" tab in the quest panel. Also, south of Varric's quarters, there is a room with a chest in it. Loot the chest to get the "Shawl of Dalesdottir" quest.
Shawl of Dalesdottir
Head to the Blooming Rose in the Red Lantern District of Hightown. Talk to Mais Dalesdottir to deliver the Shawl to her. Climb the stairs to the second floor and enter the room to the Northeast. Loot a crate there to get the Stone Toe and initiate the quest "The Paragon's Toe".
The Paragon's Toe
Head to Lowtown and find the New Surfacer. Deliver the Toe to him.
Bait and Switch
Interact with the writing desk in Gamlen's house to receive this quest. Then head to Lowtown at night (choose nighttime from the map panel as you exit the house). Then go to meet Anso the Contact in the market.
There might be hostiles here, so be prepared. Talk to Anso and agree to get back his merchandise. Then visit the smuggler's hideout in Lowtown's elven alienage at the NE part of Lowtown (still at night). Enter the abandoned house. There will be a chest in the south room, open it and then leave.
Prepare for an ambush waiting for you outside. Head back towards Anso's location. Meet Fenris, and agree to help him. Then go to the Magister's Mansion in Hightown and meet with him there. Enter the Mansion and search for Danarius. Watch out for a tough battle that includes a Rage Demon. Search all the rooms, one of them will hold a monster who has the magister's key. Once you loot the key, climb up the stairs to the Mezzanine level and unlock the door to the Magister's room. Be prepared to face an Arcane Horror. After you win the battle, loot the room and exit the house to talk with Fenris again. This will complete your quest.