Every starting area begins with a simple kill quest, has a quest to collect groundspawns, at least two distinct themes of quests, and eventually sends you twice into a nearby cave. Try to do them all, if you can. When you've finished all this stuff, you'll be given a quest of great importance to the next town over, which will serve as the focal point for the rest of your time in your first zone. (Press M to look at your map, and you'll see you haven't even scratched the surface of it.)
After a little while, and getting to level 5 or 6, it's probably time to move on. If you are a troll or orc, there might seem to be two choices, but use Razor Hill. For everyone else, it's as simple as traipsing down the road, albeit that road is mildly dangerous for dwarves and gnomes.
Your opening zone will effectively consist of four places:
There are several things you will want to do when you reach the main town for the zone:
Search for questgivers - there will be several new quests to get in the town and in the nearby areas. (If there is a nearby questgiver outside of town, there will usually be a mini-quest that sends you there, such as to Denalan, Sen'Jin Village, or Steelgrill's Depot. Don't always expect to be led to new quests, though. Explore!)
Re-set your hearthstone with the innkeeper - you probably haven't had incentive to use it yet, but your hearthstone is a valuable resource. You should try to keep it set to an area nearby to facilitate quick returns to town. Its intended purpose is to allow you to leave the game suddenly if you need to.
Consider getting started on tradeskills - you will need to train (nearly for free) the basics of a tradeskill in order to begin learning it. Not all tradeskills are going to be available to learn in the town, and you may need to go to your capital. If you talk to a guard in town, they can usually tell you if there is indeed a skinning trainer, for example. Since there is no ore anywhere on Teldrassil, a Night Elf will have to wait a bit to start on mining, blacksmithing, and engineering, but in any other case, you should consider getting started right away. If you need to go to the capital, it should probably be further down the road, but you might want to check the signposts...especially for Undead, since you'll actually need to double back from Brill.
If you want to train a weapon skill you don't have, first check (on the site) to make sure it is a legal choice for your class - no priests with guns or warriors with magic wands - and then go to your capital city to look for a trainer. Ask a guard for directions to find the weapon trainer. It may be that the weapon you want to train isn't offered in your starting city and you have to travel to another. This can be a nuisance, you may want to bear with it for now, but refer to transportation in a later section to get to the other cities.
Every capital has a bank in it. These are nearly the only places there are banks, but they all share the same bank account - items you store in Orgrimmar will be in Thunder Bluff waiting for you. You cannot store money, but there is no need or incentive to. You don't lose money when you die, and nobody can loot your stuff in PvP.
Now that you're moving on with the rest of your adventures, a few concepts to be aware of.
Experience is gained from killing creatures or completing quests. Both are effective means of gaining experience, but there is a balance to be struck between the two.
Quest experience is intended to be a bonus to balance the fact that you had to perform a specific task rather than killing any random ol' creature. Quest experience alone will not gain you levels, so getting advanced help to finish every quest in your log quickly will often leave you below the level you would expect to be from finishing an area. If you finish all the quests in your starting area and you aren't level 10 yet, you either missed some or arguably you grouped too much and will need to go "grind" some or find another zone for that level range.
Experience from creatures, aka grinding, is a fixed amount per creature, divided by the number of people in your group, and modified a little by your level compared to the target. (At level 40, you will gain no experience for a level 10 creature. At level 8, you would get a slight bonus from that target.) If there is someone in your party that is significantly higher in level, this amount will be reduced. Also, you will only gain experience equal to the amount of damage you and your groupmates do the creature. If you are struggling against a monster, and get it to half before some helpful person comes up and rescues you by killing it quickly, you will only get half the experience. Your loot will be unaffected.
Only the person who attacks a creature first will get any experience or looted. When the first damage is dealt, that creature is considered "tapped". Where you have it targeted, the color of its name will turn from red or yellow to gray. If your target has a gray name, someone snuck in a hit before you. Perhaps you didn't even see them - this happens often with hunters. Don't be disappointed when you get nothing for killing it, although you may still have to defend yourself from it!
Be considerate of other players, though. It's very rude to quickly run past a mage charging a spell just to smack the creature with your sword before the spell can finish. The number one rule to playing an MMORPG is to remember that reputation matters. People remember who you are, and you can't avoid them forever like you might on battle.net by joining another game or changing your handle.
You've probably died once by now and have already seen this, but when you die, you will get an option to "release spirit" and become a ghost. You are given this option in case someone wants to resurrect you, or your group is in the middle of fighting a boss you want credit for killing, even though you are dead. After you release, you will appear at the nearest graveyard, and you have three options:
(All deaths incur a 10% durability loss regardless of how you resurrect, but only to the things you are actually wearing. The spirit healer damages even the things in bags.)
Which leads to the next topic of durability. You might have had a diagram of a set or armor with one item in yellow or maybe red. This indicates that item has low durability. When the item reaches zero, the item is broken and will not work. It can be repaired and be fully functional again.
You can restore items to max durability by going to a vendor who sells some sort of weapons or armor, and selecting the repair option. At lower levels, this is a trivial cost, but it scales up with your equipment. However, if you use the spirit healer, which completely breaks all your equipment, including those in your bag, this cost might be more than you can afford.
Eventually you will finish all the quests in your newbie zone and have to say goodbye.