There are a great many places to see in the world and many things to do, and getting to each of them requires a bit of hiking. Fortunately, there are faster ways to get somewhere than simply hoofing it through dangerous territory.
The first and most obvious thing you will need to know is how to get where you are going. Especially where anywhere within a zone is concerned, pressing M to see your Map is helpful for navigating. On the area map, right-clicking will reveal the entire continent or using the dropdown arrows will allow you to select a different zone for longer-distance travel.
Of course, one of the most important aspects of travel is Discovery. Until you have visited a point of interest, it will not appear on your map. It's also fairly likely you will have to walk somewhere the first time you want to reach it.
Walking everywhere constantly would be an incredible drag, which is where Flight Masters come in. Any new Flight Master you encounter will have a green exclamation point over his head and on the minimap to indicate you have not acquired this Flight Point.
Once you have received credit for the flight point, you may travel to neighboring flight points for a small fee. If you are traveling somewhere more distant, your character might have to fly to another location in between, but the flight will continue non-stop. Unfortunately, if you have not yet discovered the flight points that connect to your location, you won't be able to fly anywhere!
During flight, your character will be transported quickly, but not instantly, to your destination. It is completely safe to step AFK while you fly. Each faction has their own flight network for each continent.
Note: You have to "discover" each of the Flight Masters in order to travel to them.
There are several boats and zeppelins that link together Kalimdor and the Eastern Kingdoms. Boats (and zeppelins) go between two specific locations, and will pause at each end of the route for a long enough period for players to embark and disembark. After that, the boat will begin to travel on its own to the other stop.
There are boats or zeppelins between:
There is also a boat that connects Auberdine to the Night Elf tree-island of Teldrassil, and a boat that runs from the mainland to Feathermoon Stronghold in Feralas, neither of which involve changing continents.
Mounts are a much swifter means of travelling on foot. Although each race has a different mount, the basic mounts all travel at the same speed. Upon reaching level 30, a character can learn to ride the mount of his race (or another race he has reached exalted reputation with.) Combined with the actual purchase of the mount, this has a cost of 100 gold, minus discounts. Warlocks and Paladins also have a special class mount.
Mounts can be summoned any time your character is outdoors and out of combat. A standard mount will increase your character's movement speed by +60%. At level 60, the character is elgible to learn to ride a swift mount, often called an epic mount, which costs considerably more and travels at +100% movement speed.
Players who reach level 70 can also learn to use a flying mount. Players must have their swift mount first to do so. There are two speeds of normal flying mounts, a 1000-gold basic flying mount that travels at the speed of a basic mount, and a 5000-gold epic flier that moves at the speed of a swift mount on the ground and at +280% speed while in the air.
In addition to this druids learn flight forms. The first of which they learn at level 68 simply called Flight Form and does not require the 1000 gold fee to learn. The Swift Flight Form requires that you have artisan riding skill (5000 gold skill) and you must go through a rather long quest chain to attain it.
Flying mounts may only be summoned in Outland.
Through the art of magic, there are some more expedient ways to travel. The most widely-used of these is the hearthstone, which immediately teleports a player to their home point. The shaman ability Astral Recall performs the same function without putting the hearthstone on cooldown.
Mages can travel to other specific locations by casting Portals. Beginning at level 20, they begin to learn the means of teleporting themselves to certain major cities, and at level 40 they can learn to create an actual portal that other players can use. A few select portals like these exist in other locations, notably in the center of the city of Shattrath.
It is also possible to summon a player to you. A warlock can perform a Ritual of Summoning if there are two other group members present to participate. This allows a target (fourth) group member to be brought immediately to that location. This process can also be performed at any Meeting Stone by any class, and only requires one other groupmate instead of two.