Flight paths are what most players will use to travel between nearby zones. You must discover a flight path before using it for the first time--these are marked in green on your minimap.
We've got a page covering Riding: skill levels, mount types, speeds, and training costs. You can start off riding ground mounts at level 20, increasing your travel speed and learning how to fly at later levels.
Mages can port players to capital cities, as well as teleporting themselves.
A warlock's Ritual of Summoning is useful for straggler players left behind--and especially getting a low-level player to an otherwise inaccessible zone.
Summoning stones are also found outside major instances and raids. This summons individual players to the stone's location.
With Recruit a Friend, you can summon linked characters to each other's locations anywhere in the world. Recruit a Friend is great for quickly leveling characters, as there are triple experience and increased reputation gains, as well as bonus levels that can be granted to characters.
Many capital cities have portals to other capital cities--Stormwind, Orgrimmar, and the Shrines of Vale of Eternal Blossoms are particularly good.
There are many varied items that can port you to special locations, like guild capes and teleport tabards. The "Getting To:" sections of this guide include a map with mouseover travel details, as well as tabs describing every item and method of transportation, ranging from simple town portals to elaborate portals that can only be accessed after completing quest chains.