Okay, even though Denerim is like Loghain's home base, you can visit it early to start and finish some quests.
Back Alley Justice 4:53
You can accept on the Chanter's Board. This quest has you kill 3 different gangs in the outskirts of Denerim. Afterwards, you can report to the Chanter and Sergeant Kylon for monetary reward.
Blood of Warning 10:20
You can accept more quests from the Mages' Collective Bag. Blood of Warning is completed in Denerim by interacting with four specific doors. The first two are on the west edge of the Denerim Market District (15:24). Another is in the Dark Alley location accessed by the Denerim City Map. And the last is in the Dirty Back Alley.
10:40 Sergeant Kylon
Kylon will offer you two quests. The first, Peals Before Swine, has you disband a rowdy group of Mercenaries at the Pearl; Kylon will only reward you for the job if no Mercenaries are killed at the Pearl. When you attempt to return to the Market District to report to Kylon, he'll intercept you to give your pay, but you and he will be ambushed by the Mercenaries. Take them out this time and loot Aodh from Cristof's corpse. Afterwards, speak with Kylon for your reward and congratulations.
If this quest resolves with you having correctly followed his orders, Kylon will offer the Crimson Oars quest back in Denerim. Like Pearls Before Swine, you'll have to drive out some Mercenaries. You'll find them in the Gnawed Noble Tavern in the Market District. You are allowed to kill them this time. After dealing with them, report back for your reward.
Additionally, if you complete Back Alley Justice, you can speak with Kylon for a bonus reward of gold.
Trial of Crows 18:58
In the market, after you have met Zevran, you can speak with Master Ignacio. The next time you try to leave the Market to access the rest of the city, a messenger boy will intercept you to deliver a letter from Ignacio. He wants to speak with you in the Gnawed Noble Tavern about working for the Crows. Each completed contract nets some pay and aids the Warden's cause.
The first task is to kill four ambushers in the Pearl who have laid a trap for Warden sympathizers; (you can complete this before ever meeting Ignacio, and he will still reward you for it). To meet the attackers, you'll need to inspect the Grey Warden poster on the east wall of the Market District in order to receive the door password.
The second task involves two separate contracts. One contract has you killing Kadan-Fe Qunari mercenaries. This is easy; just visit their camp marked on your World Map. The other has you killing an ambassador in Orzammar. The ambassador can't be killed until you have access to the private quarters in the Dwarven Royal Palace, (which can be gained by completing Paragon of Her Kind).
The last contract has you kill kidnappers. Simple; you go to the meeting spot and kill the guys.
Your reward for all this is some unique gear and a vote in favor of the Wardens at the Landsmeet later in the game.
Favors for Certain Interested Parties 23:30
At the Gnawed Noble Tavern, you can accept quests from the bartender. Most of them are simple, but Sign of Safe Passage requires the Warden to equip a short or long bow, so keep one that you can equip even if you're a mage.
Blackstone Irregular Box 24:32
More Blackstone quests are available. Restocking the Guild can be completed immediately. Speak with the liason, and he will take the 20 of your weakest health poultices for the quest to complete.
Brother Genitivi's Home 26:45
You'll need to visit this home to start or advance the Urn of Sacred Ashes quest. Even in the home, you'll meet an imposter Weylon. Either can get him to attack you, or try to inspect the back room.
After Weylon is dead, loot the chest in the back room to learn that the village of Haven is linked to Urn.
Wonders of Thedas 30:30
At this shop, you can buy the Staff of the Magister Lords and Reapers' Vestments, two of the best pieces of gear for Mages in DA:O.