Potions, Portals, and Scrolls of Recall: How to get around Azeroth as quickly as possible
Azeroth is big. This is readily apparent to any new player who's confined to running around on foot or anyone too cheap to shell out for a mount. Even mounted, you're going to spend a lot of time getting from point A to point B unless you can somehow shorten the journey -- and you can usually shorten it in a number of interesting ways.
We all know about the zeppelins and the boats, the workhorse transportation system of Azeroth, but due to the eccentricity of the many transport options in the game, the shortest distance between two points is not necessarily a straight line. For example, an inventive Horde player without a Hearthstone up who wants to travel from Silithus to Orgrimmar can settle in for a long flight -- or she can simply chug a Potion of Deepholm and take the Orgrimmar portal from the Temple of Earth. No potion, but you've quested through Sholazar Basin? Take the Titan Waygate from northern Un'Goro Crater to Sholazar, then fly south to the Horde's zeppelin at Warsong Hold. You'll still beat a wyvern flying from Cenarion Hold.
With even a few of the following options, a max-level player with some imagination should be able to scrape together a few methods of considerably shortening a journey.
Class-specific travel abilities
Many classes have portal or speed abilities that can transport them across continents or at least shorten a journey somewhat.
Note that only mages and warlocks can transport players other than themselves.
Death knight The Death Gate spell learned at the end of the death knight starting experience will transport you to Acherus above the easternmost portion of the Eastern Plaguelands. From there, you can catch a flight to anywhere in the Eastern Kingdoms.
Druid The Teleport: Moonglade spell puts you in front of the house at the extreme northeastern end of the Nighthaven village in Moonglade. Druids also have access to special, no-cost flight paths to Darnassus (Alliance) or Thunder Bluff (Horde) departing from 44, 45 (the extreme southwestern end of Nighthaven). This is actually a relic of the old quest for Bear Form. Travel Form is also available as a personal speed boost.
Hunter While hunters have Aspect of the Cheetah to increase their run speed by 30% and Aspect of the Pack to increase their group's by the same amount, they do not have any instant transport abilities.
Mage Oh, boy -- this is the class to play for anyone who's too impatient to wait for a zeppelin or ship. A level-capped mage will be able to teleport to all four of his faction's major cities (Stormwind, Ironforge, Darnassus, and the Exodar for Alliance; Orgrimmar, Undercity, Thunder Bluff, and Silvermoon for the Horde). He will also be able to teleport to a secondary city (Theramore for the Alliance; Stonard for the Horde) and, later, the neutral cities Shattrath and Dalaran and the not-so-neutral territory of Tol Barad. Not only can he teleport himself, but he can also portal other players to all of these places -- pretty sweet, but it'll cost you. Each teleport requires a Rune of Teleportation, and each portal requires a Rune of Portals. (Both can be purchased from reagent vendors located in all towns and cities.) If you're asking a mage for a port somewhere, it's considered polite to buy your own rune in addition to a tip. If you're a level-capped mage with all teleport and portal spells trained, you're an engineer, and you've picked up some of the options available from this article, you basically have no excuse for not being able to be anywhere in Azeroth in under 2 minutes.
Paladin While paladins have Crusader Aura to increase their (and their parties') mounted speed by 20%, they do not have any instant transport abilities.
Priest Priests do not have any speed or instant transport abilities, unless you count being a jerk to people with Leap of Faith.
Rogue Rogues do not have any speed or instant transport abilities unless you count Sprint. However, the subtlety rogue's Shadowstep can occasionally be put to creative use.
Shaman The shaman has Astral Recall in order to teleport to her hearthstone location every 15 minutes. The Glyph of Astral Recall can shorten this to a 7.5-minute cooldown. Ghost Wolf is also available as a personal speed boost.
Warlock Warlocks are unusual in that they (in tandem with two other players) can create what is effectively a meeting stone anywhere in the world with the Ritual of Summoning. This spell is trained at level 42. However, warlocks do not have any speed or instant transport abilities.
Warrior Warriors do not have any speed or instant transport abilities, unless you count Intercept's being a giant pain in the ass and Heroic Leap's use as a pseudo-Blink.
Gadgets, tabards, and gear
These aren't constant travel items. All of them have to be created, won, or bought and/or have cooldowns associated with them, but a bag with at least a few of them will give you some quick options.
Argent Crusader's Tabard This is a rather attractive tabard that can port you to the western pavilion at the Argent Tournament every 30 minutes, but you'll have to put some time in at the Tournament to be able to buy it -- also, 50 Champion's Seals.
Atiesh, Greatstaff of the Guardian Very few players have this, but if you do, you can create a portal to Karazhan with it with basically no cooldown. I would love for Atiesh, or at least its model, to be back in the game.
Baradin's Wardens Tabard (Alliance) / Hellscream's Reach Tabard (Horde) You'll have to get to honored with the respective factions in order to purchase this, but fortunately that doesn't take too long. It ports you to Tol Barad on a four-hour cooldown, and from there, you can take a port to Stormwind or Orgrimmar.
Blessed Medallion of Karabor This will port you outside the Black Temple in Outland's Shadowmoon Valley (about a skip away from the meeting stone) once every 15 minutes. It's highly useful for anyone who wants instant access to the Outland continent, but you'll only get it at the end of a long and involved quest chain that culminates in an Illidan kill.
Boots of the Bay This was added in late Wrath as one of two potential rewards from winning the Kalu'ak Fishing Derby on Saturdays (the other reward is of course the heirloom Dread Pirate Ring). The cooldown is incredibly long, but it's the only way to get instant access to the southern part of the Eastern Kingdoms without being hearthstoned there.
Direbrew's Remote This is one of the more famous options and deservedly so, as it's pretty cool. However, it drops only during Brewfest from the Keg-Shaped Treasure Chest you'll get once per day from Coren Direbrew kills. Using it afterwards will give you a Dark Iron mole machine that will take you to the Grim Guzzler bar in Blackrock Depths. Oh, and riding the machine there will also give you a free concert from The Artists Formerly Known as Level 80 Elite Tauren Chieftain when you talk to their goblin roadies.
Jaina's Locket You'll only have a shot at this if you're in a raid that kills the Lich King with a Shadowmourne present. When that happens, an Unsealed Chest will drop with several items inside to be returned to five central lore figures. The reward from the quest associated with Jaina's locket is -- what else? -- a locket that can create a portal to Dalaran once an hour.
Potion of Deepholm This can be created by alchemists with 525 skill, but it's usable by everyone. This will send you straight to the Temple of Earth in the center of Deepholm, which itself contains portals to Stormwind and Orgrimmar.
Ring of the Kirin Tor I've linked the achievement here rather than the rings themselves, as there are four different versions. However, even at exalted they'll run you 6,800 gold, so they're not an idle purchase. From Dalaran, you can take a portal to Stormwind or Orgrimmar.
Shroud of Cooperation (Alliance)/Shroud of Cooperation (Horde), Wrap of Unity (Alliance)/Wrap of Unity (Horde), and Cloak of Coordination (Alliance)/Cloak of Coordination (Horde) These three cloaks are sold by guild vendors and can teleport you to Stormwind or Orgrimmar. You'll need to reach honored with your guild to be able to purchase them.
One-way portals
The obvious disadvantage of one-way portals is that you can't immediately return to your point of origin. Be sure that the destination is where you want to be before you take it, because there's no save button in WoW.
Major cities to Blasted Lands I can't believe I initially forgot this one, but every major city has a portal (always located close by the city's portal trainer) to the pathway just outside the Dark Portal in the Blasted Lands. This is an easy way to reach Outland. You can technically convert this to a two-way portal (see below) if you go through and then take the ports back to Stormwind/Orgrimmar.
Shattrath to Tanaris This isn't actually a portal but rather a bronze dragon NPC named Zephyr in High Elf form levitating in the World's End Tavern in Shattrath. You have to be revered with the Keepers of Time in order to take the port. If so, she will teleport you to the entrance of the Caverns of Time in eastern Tanaris.
Shattrath to the Isle of Quel'danas There's a portal located in the northwestern portion of Shattrath's central building (where A'dal floats) that will take you to the top of the Sun's Reach Sanctum. From the Sanctum, you can grab a flight elsewhere in the Eastern Kingdoms or do a quest that will grant you a teleport back to Shattrath (see below).
Dalaran to Tanaris This is another port directly to the entrance of the Caverns of Time, but in this case it's an actual portal and not an NPC. You'll find it slightly to the left up the top of the stairs in Dalaran's Violet Citadel in the southwestern part of the city.
Stormwind/Orgrimmar to Vashj'ir A Cataclysm-era portal located in the shamanic circle of both cities. There is not a port back.
Stormwind/Orgrimmar to Uldum A Cataclysm-era portal located in the shamanic circle of both cities. There is not a port back.
Two-way portals
While these are rare, they're valuable. You can go back and forth between them to your heart's content with no cooldown.
Titan Waygate: Un'Goro to Sholazar Basin If I recall correctly, this has to be activated by way of a quest line in Sholazar (beginning with Force of Nature, given by Pilot Vic in River's Heart), but once it's up and running, you can use it permanently.
Darnassus and the Exodar While this is Alliance-only, it will allow you to move between the two cities without having to take multiple ferries.
Stormwind/Orgrimmar to Deepholm While these technically aren't two-way portals, the set of one-way portals between each zone are sufficiently close to qualify. You'll find portals to Deepholm from Stormwind or Orgrimmar located in each city's shamanic circle. Portals back are located in the Temple of Earth where you zone in.
Stormwind/Orgrimmar to Mount Hyjal A Cataclysm-era portal located in the shamanic circle of both cities. Portals back to Stormwind and Orgrimmar are located on either side of the entrance to the building where you appear.
Stormwind/Orgrimmar to Twilight Highlands Another Cataclysm-era portal.
Tol Barad to Stormwind/Orgrimmar Each faction's encampment on the Tol Barad peninsula has portals to their respective capital cities.
Profession-specific travel options
When I say profession-specific, what I largely mean is engineering.
Dimensional Ripper: Everlook This can only be built and used by Goblin engineers with a skill of 260 or above. One of the very few travel options, apart from the druid's Moonglade port, that can put you on the northern end of Kalimdor. Four-hour cooldown.
Dimensional Ripper: Area 52 This can only be built and used by Goblin engineers with a skill of 350 or above. This is slightly less convenient to Shattrath than the Gnomish port to Toshley's Station, but the difference is minor. Four-hour cooldown.
Ultrasafe Transporter: Gadgetzan This can only be built and used by Gnomish engineers with a skill of 260 or above. This is less useful than it once was, given the number of portals now available to the Caverns of Time, but eh. Four-hour cooldown.
Ultrasafe Transporter: Toshley's Station This can only be built and used by Gnomish engineers with a skill of 350 or above. Horde players, don't worry! This actually transports you outside of Toshley's Station proper, and you won't be attacked by the guards when you port in. Four-hour cooldown.
Wormhole Generator: Northrend This is one of engineering's more interesting and random toys. Rather than porting you to one specific location like the other transport tools, this will dump you in one of six locations around Northrend: Above the small central lake in Borean Tundra, on a small ledge toward the top of Utgarde Keep in Howling Fjord, on the Glimmering Pillar in Sholazar Basin, in the mountain range southwest of the Argent Tournament in Icecrown, and outside of Ulduar in Storm Peaks. Very occasionally, you'll be teleported to a small chamber in the Dalaran sewers, and you'll be able to talk to a Forsaken NPC, Kaye Toogie, who can sell you the schematics for engineering's dragonling pets. Beware: You'll have to use another port option to get out.
Scrolls of Recall These are scribe-only, but they'll teleport you to your hearthstone location without triggering your hearthstone cooldown.
Hearthstones and miscellaneous options
If you lose or accidentally destroy your hearthstone, just speak to any friendly innkeeper for a new one.
Hearthstone (and hearthstone doppelgangers) This is the first option you'll get for teleportation, and it's often the option of last resort. The Innkeeper's Daughter from archaeology, the Ruby Slippers from Karazhan, and the Ethereal Portal from the WoW Trading Card Game all share its cooldown, so ... they're kinda useless, sad to say.
Last Relic of Argus This, hands down, is my favorite archaeology item. It's hugely unpredictable, but if you're in a gambling sort of mood (or if a world PvP situation's gone south in unfriendly territory and you need to get the hell out of Dodge), this can send you to one of 25 different places on any continent barring Northrend. Locations are as diverse as the Duskwood catacombs and Fray Island off the coast of the Northern Barrens. More stuff like this in archaeology, please!
Isle of Quel'danas to Shattrath The daily quest Know Your Ley Lines will reward Darnarian's Scroll of Teleportation, which will port you to Shattrath. It only works from the Isle, though, so its use is limited. Back during The Burning Crusade, it was the usual method for players to return to Shattrath after finishing their Sunwell dailies.
Trans-Dimensional Fabricator Now this one's a bit weird. Apparently it was never in the game at all, but its related spell -- "Teleport with error" -- is the one used by the Last Relic of Argus. Was the Relic originally supposed to be an engineering item that made its way to archaeology?