Quest hunter with plenty of synergy with its 1-cost minions.

Playing the quest on turn 1 prevents you from snowballing in the early game in most cases, so you are looking to simply keep up as best you can until you have 4 mana available. Questing Explorer and Tar Creeper are the best cards to look for to help you do this, but your 1-cost minions can help as well while progressing the quest. Tracking helps a lot to set up the midgame combos below.

Once you have 4 mana, you can start to attempt to come back or create a lead by using Hobgoblin or Magic Carpet aling with a 1-cost minion. Springpaw is generally the strongest 1-drop to receive a +2/+2 as it can immediately make use of the stats, while Argent Squire is the better carpet buff receiver.

At 5+ mana, Toxmonger Combos become available, with Elven Archer and Kobold Lackey being the standout ones since they immediately eliminate an enemy while leaving a poisonous minion behind.

Along with springpaws, Fire Fly, EVIL Cable Rat, and Halazzi, the Lynx are all useful to progress the quest in a timely manner, while being reasonable cards in their own right.

In the lategame, we have Queen Carnassa, who shuffles in Carnassa's Brood that you can give charge with a Tundra Rhino, or use with Magic Carpet/Toxmonger. Finally, once the opponent is low enough, we have Kill Command.

Other cards I'm considering include:

Jar Dealer to further accelerate quest completion.

Dire Mole is an option over Argent Squire when you are against decks that play a lot of 1/1s that also has the benefit of being another cheap beast.

Glacial Shard for a 1-drop that always stays relevant, and the extra point in attack makes it a good target for Magic Carpet or Goblin Lackey buffs.

Deathstalker Rexxar to generate infinite threats vs control.