For the "Under the Shadow" quest you need to collect 10 Necrotic Runes. These runes are an uncommon item (though they are very common in drop rate) which you can get from killing any undead / scourge mob around a necropolis spawn point. You can find the zones these necropolises are in by looking for a purple skull on your continent map. Be forewarned though, there's a lot of people farming these items and you'll be hard pressed to get any without a group.
The "Shadows of Doom" quest requires you to kill an elite mob, a Shadow of Doom. These mobs don't hit very hard but are rather difficult to get a hold of unless you're in a group with a plan. In order to make a Shadow of Doom mob appear, you need to give 8 Necrotic Runes to a priest NPC near a necropolis spawn point – they're in the same area you'll pick up the Necrotic Runes from. To complete the Shadows of Doom quest, you'll only need to kill one Shadow of Doom.
Once the quests are complete (although only "Under the Shadow" is required to access the Quartermaster), turn them back in to Commander Thomas Helleran. Celebrate as you've gotten either some Blessed Wizard Oil or some Consecrated Sharpening Stones. But the next part is even more fun. Spend all those extra runes you've picked up, or go off and farm more for a few hours. You can spend them at the Argent Dawn Quartermaster who is located off to the left of Commander Thomas Helleran.
Some of the big items that the Argent Dawn Quartermaster provides are the Argent War Horn, which summons an Argent Dawn Knight to defend you (you can see a video of it now, and we'll be having a post up about the Kngiht shortly).
Another cool item is the Argent Dawn Banner, which plants a banner on the ground and spits out a consecrate. You'd think it's a useful item for tanking, but it really doesn't add anything – especially given the buffs tanks received in patch 3.0.2. But having a banner planted at your feet is cool.
Also provided is a Tabard of the Argent Dawn. This is a nifty tabard that lets you look like all the Argent Dawn NPCs around you. Once you get the Tabard you also complete a "Feat of Strength" achievement called "Tabard of the Argent Dawn".
Finally there is the Argent Tome which smites your foes with your wisdom. Being that you're wisdom is full of truthiness, it is clearly an over powered trinket.
There are lots of other goodies, including mana and health potions, as well as some pieces of armor you can get. Take a look at our gallery for everything available from the Argent Dawn Quartermaster.