My personal recommendations for absolute beginners with little confidence:

Culling of Strathholme:
Ignore the clock completely and just slowly work through the instance. The bosses are all basically tank and spank with little twists. Heal whoever gets chained on the first boss, heal whoever gets blown up on the second, aoe heal a bit when the third boss berserks, and face the last boss away from the group/drop a grounding totem if you have one. Grats on an epic, decent loot, and a frozen orb.

Drak'theron keep:
First boss is just a tank and spank with a bit of a timer. Aoe heals really help. You can skip the second and even the third bosses if they give you any trouble. The final boss is the easiest final boss of all the heroics. Grats on an epic, decent loot, and a frozen orb.

Gundrak: You can do Moorabi and the Snake guy pretty easily, and they're only 3 pulls each before you can attempt them. If you have an aoe healer the golem/elemental is a joke. Finally the last guy is very easy. Just keep him at range when he's a troll, hug him when he's a rhino, and clump your guys in a corner.

These three instances are extremely easy. However once you figure out the little tricks for each heroic, they're all pretty easy. It's simply a matter of running them enough times and reading up on the boss fights that you figure out the easiest/best way to do it for you. There is a lot of absolutely great information in the stickies.

Once you're at the level where any heroic is no problem, you'll find that the two easiest to farm for badges/orbs/abyss crystals are violet hold and culling of strathholme. They're both more or less located in Dalaran, and you can do them both in less than an hour for 7 badges, 2 orbs, 2 abyss crystals, and a ton of other de/sellables.