What is the fastest way to move upward?

The fastest way to travel upwards is on an upward slope of powered minecart rails.

Update

Tell a lie, there's actually a faster way, our good friend TNT. Use this at your own risk!

Pistons elevators are the fastest way to go in 1.7.3 and above.

I believe minecarts would still be the fastest, provided that you have enough space and resources for them.

Stairs would probably be faster than ladders, but again require more space.

Boat elevators and water ladders are now fixed entirely. They will not work.

Ladders are pretty decent. Slow-ish, but efficient.

If you had a fast PC, you may be able to fiddle with portals and see if you can get them working such that you enter portal B at the bottom, arrive in the Nether, get back in the portal, and come out in portal A at the top. For long distances, this is probably pretty fast..if a little unorthodox.

If you don't consider it cheating, WorldEdit's ascend/descend commands are instantaneous and using Single Player Commands can be bound to any key.

If that's too cheated, there's also an elevator mod (also for SMP if you use bukkit: link).

If you insist on vanilla minecraft, fredley is probably right, you'll have to use a powered minecart or TNT - I guess with enough sand blocks between you and the TNT directly below you, you'll even survive being catapulted up, and maybe use some trapdoor mechanism wired to the TNT detonator to shut just in time so you stay upstairs...

edit as akaGrim linked to this youTube video, a minecart elevator might be quicker and still "legal".

As of 1.6.6 it appears boat elevators work again. http://notch.tumblr.com/post/6035819427/minecraft-beta-1-6-6

The fastest way to move upwards (faster than TNT!) is the /fly command, when used with an argument for speed.

/fly 255

...will have you so far above your map you can't even see it within a second or so.

This only works in SMP, or with Single Player Commands installed.

I saw something on youtube about "teleportation". It isn't really teleportation. It's more like just really fast travel. The basic idea is to cram a lot of mobs into a small space then move the player next to them. The combined "bump" of most of the mobs will add up, propelling the player very quickly over long distances.

I'm not certain that combined bumps can be used for vertical travel, but I would bet they could (maybe bump into stairs or get bumped from below?). The only problem is that this device, while it works, reduces my framerate to less than one per second, so I can't investigate vertical movement using this method.

It would be interesting if someone could take that guy's idea (I forget his name) and turn it into a vertically-moving "teleportation" machine. I simply don't have the computer power for it, though.

As of 1.13, tridents + pearls provide a fantastic method of quickly launching yourself upwards, by using the momentum of the trident to speed up the pearl. This is usually achieved by holding the trident in your off-hand, then selecting the pearl in your hotbar. You do need something to hit with the pearl, though.