This guide will give you a step by step path to complete all the Cataclysm dailies in as little time as possible using Deepholm as the starting point. You'll want to do this to gain reputation for the new factions. Not all new factions have daily quests so for those that don't, you'll want to buy and equip a tabard and queue for the new dungeons.
Set your hearth to the Temple of Earth at the World Pillar in Deepholm or buy a Potion of Deepholm for this great speed run. Travel from Temple of Earth -> Therazane -> Temple of Earth -> Stormwind -> Uldum -> Tol Barad -> Twilight Highlands -> Stormwind -> Bonus Archaeology around Azeroth.
Therazane Rep in Deepholm (Six Quests or Seven if Revered)
First, go to Therazane's throne using the portal on the north end of Temple of Earth and grab the five dailies there. Then head east to the Goreworm hole. Quickly go inside to pick up the quest from the goblin there. Head into the cave and grab the ores, only killing the Goreworm if you need to get by them. The goal here is to get all the ore and get out of the cave. You'll find two types of spikey type ores and two types of block type ores to explode, as well as some crystals lying around. Once you have those done head back to the entrance and turn in the quest to the Goblin. If you have any Goreworms left, complete those outside in the surrounding area and to the south as it's a lot easier. (Note, this Goblin quest also rotates with two other quests that you would pick up rom the Therazane throne. They all are in the Crumbling Depths, though.)
Next, you'll want to head to the southeast and up to the fungal growth area. Here you'll want to destroy the little shrooms. Sometimes they explode and knock you pretty far so be careful not to fall off (Druids need not worry because of their flight form can prevent their fall!). The mushrooms sometimes give a buff to run faster, so do those first. Bounce between mushrooms and occasionally kill the Fungal Lurkers. You'll need five of those. Make your way northeast towards the pool to pick up some miracle grow. Once you're done, head back to Therazane's thrown to turn in the remaining five dailies. If you're revered you can do a sixth quest called The Restless Brood to the west. From there, take the portal back to the Temple of Earth and then catch the port to Stormwind. View the Therazane rep rewards.
Cooking and Fishing Quests in Stormwind (Two Quests)
When in Stormwind, you'll want to do both the cooking and fishing dailies available near the entrance to the Stockades. These send you around the city and are generally pretty quick to do. Learn more about Cataclysm profession rewards.
The Ramkahen Dailies in Uldum (Two Quests)
After you're done with those two head to the north end of the city to the Earthen Ring portal area and take the port to Uldum. There's one daily to the east that you can do for another +150 reputation with Ramkahen. It's
Daily for the Venomhide Ravasaur Mount (One Bonus Quest, Horde Only)
If you're horde and doing the epic raptor mount quest you're now be close by to Un'goro and Tanaris so you'll want to quick do your daily for the little raptor.
Baradin's Wardens on the Island of Tol Barad (Six Quests on Peninsula, Five Quests on South Island )
The next set of dailies to do are in Tol'barad. If there isn't a battle going on you can pick those up. Bring a buddy if you can as the area is pretty dangerous. If you're an herbalist there are lost of Azshara's Veil in the rivers and also the cove (beware the elite shark there!). It helps to bring a potion of Water Walking if you have a quest to go to the cove. Once you finish those quests on the prison island complex head back to your base camp to take the port back to Stormwind. View the Baradin rep rewards.
Wildhammer Reputation in Twilight Highlands (Five Quests)
You're ready to do the final set of dailies, which are Wildhammer one's available from Thundermar after you complete the dwarf quests leading up to the wedding. There's another five quests you can do there and there are some really nice rewards. Once you're done with those, you can use your hearthstone to head back to the Temple of Earth. View the Wildhammer rep rewards.
Bonus! One More Portal for Archaeology Dig-sites
Now that you're back next to the ports in Stormwind you might want to check out your map for archaeology dig-sites and cool down with some relaxing sightseeing. Sometimes you can port pretty close from the Twilight Highlands or Hygal to get sites close by, but keep in mind if you just finished Wildhammer dailies you'll be in northern Eastern Kingdoms already.. There are some really great epic rewards for this profession so be sure to give this profession consideration for your daily quest runs. Read a guide for the Archaeology profession.
So there you have it! That's the fastest way to do the set of dailies available in Cataclysm. I hope you found this helpful.