As far as general tips, don't be in a rush. The level cap is about to be raised to 80 and, frankly, you can get to 70 without doing half the quests, which is a shame, since the quests are the best part of the game.
Do all the quests in your racial homeland. There actually is a story in each of these regions, and it'd be a shame to miss out on it. The Tauren, undead, dwarves, human, draenei and blood elves have particularly strong storylines, but everyone but gnomes and trolls have good ones.
Do as many instances as you can, as early as you can.
Turn off General. Just doing this will add years to your life and make the game much more enjoyable.
PvP at least a little. The Horde/Alliance are dirty villains and most are deserving only of death!
Once you reach the Outland levels, try and get Shatar, Honor Hold/Thrallmar, Cenarion Expedition, Lower City and Keepers of Time at least to Honored, as fast as you can. This gives you the heroic keys for the level 70 versions of dungeons and gets you a starting set of PvP armor you can buy -- going with non-PvP gear is a rookie mistake that you will regret.
If you're using Auctioneer, don't bother being a farmer. Do at least one item-creating profession and probably the associated gathering skill. Pick one that suits your character and is fun for you. For me, mining and making my hunter's guns, scopes and goggles are as fun as any dungeon or raid.
Raid at least a little. Many servers, especially the RP ones, have a raiding alliance system where you can participate in raids without your guild. At lot of the most interesting content can only be seen this way.
Oh, and play a shortie. The world is much better as a shortie.