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In addition to the travel options below, setting your hearthstone to a good place is key--a location that's not easily accessible through other means. For example, Alliance players pushing for Tol'vir digsites may want to set their hearth to Desolace since there's easy ways of getting to other corners of the map via Darnassus, Theramore, Uldum, and Hyjal.
All players can get a new hearthstone and change the location by talking to any innkeeper.
Hearthstones are on a 30 minute cooldown--it used to be an hour before Cataclysm. It can be reduced to 15 minutes with Hasty Hearth, the Level 8 Guild Perk.
There are some fancy versions of hearthstones:
The Innkeeper's Daughter is a Dwarven Archaeology solve that tells a tale about the darker side of Dwarven culture.
Ethereal Portal is a rare item from the Trading Card Game--when you hearth, a neon portal in the style of the Consortium Ethereals pops up.
Dark Portal is another rare item from the Trading Card Game--it creates a mini-version of the Dark Portal.
Ruby Slippers, a drop from the Opera event in Karazhan, are a clear reference to Dorothy's sparkling red shoes in the Wizard of Oz. They're boots with a hearthstone built in! These four items all share a cooldown.
The Last Relic of Argus is the anti-hearthstone; on a 12 hour cooldown, it will port you to an obscure place--and generally in an odd position, like on a tree branch. However, if you're bored, let the Last Relic port you to an area and see what's around to do. Doesn't share a cooldown with Hearthstones, and has the added bonus of being a 5 second cast affected by haste.
Scroll of Recall, Scroll of Recall II, and Scroll of Recall III are also unpredictable. They port you to your Hearthstone's location, but don't share a cooldown--which is great. However, as you level up, the results become more unpredictable, so it's a bit of a gamble to use them.
Shamans get Astral Recall, which is an additional hearth on a 15 minute cooldown--7.5 when glyphed with (Spell #58058). This spell is affected by haste and can be cast with Spiritwalker's Grace--meaning a mobile hearth!
You get a special additional Hearthstone at your Garrison at level 90+ in Draenor, as well as Admiral's Compass to the Shipyard for completing 50 Missions.