How to move faster in Terraria?

There's rocket boots you can get for going up faster.

Each Anklet of the Wind you equip adds 10% to movement speed. (But as of 1.06, they no longer stack, though you should be able to have one Anklet and one Aglet)

Hermes Boots increase movement speed by 100%, but take a while to accelerate.

There are multiple ways to go fast in Terraria as for the 1.2.4.1 version.
I'll list all the non-accessories ways to transport yourself, many of them imply blocks or wire placement.

Now, in 1.3, if you have a stopwatch in your inventory, you can know precisely your speed at any given moment. Example : Your normal walking speed without any bonus is 15 mph.

Pre Hardmode Tier

Sunflowers

Gives you the 'happy' buff, which always works as long as at least one sunflower is in your proximity. Percentage seems to be 20% or 25%.

Swiftness potion & food

+45% speed
the potion add 25% to your speed, and the food (which gives you the 'well fed' buff) will add 20%.

Note : These speed buffs only affect your max speed without any speed boots on.
If you have them, these buffs will only affect your acceleration and not increase the max speed.

Ropes

Very cheap yet very useful. With them, you can travel long vertical distances without caring about fall damage, you can also travel horizontally, but it's very slow.

• upward speed : 41 mph
• downward speed : 51 mph (the exact same as regular falling speed)

Speed Hookshot

If you are pretty familiar with the hookshot, you can move around quite quickly

Horizontal : Hook the ground to the far left/right and jump before it has fully retracted to keep the speed. Flat ground, and up-slopes are recommended for this use.

Vertical - upwards : set some platforms, on the way up, and when you hook them, jump just before reaching them to keep the vertical speed and re-use immediately the hookshot to reach the next platform.
However, if you have ropes, you better use them on vertical pathways.

• normal hookshot : 55 mph
• bat hookshot : 81 mph
  its speed can be overwhelming sometimes.

Rails & Minecarts

you can go horizontally or diagonally with a very solid speed. All you need is some iron and wood.
1 wood + 1 iron bar (or 1 lead bar) = 50 pieces of track.

You can make the minecart jump with a slope in order to spare some tracks blocks, make sure your speed is sufficient and your track planning is correct, or you might suffer fall damage inside the cart.
You can even change direction thanks to bumper tracks (hammer the end of a track twice)

Now, with the 3 different carts, universal track jumping is not recommended, as the max speed is different from the 3 carts (but it can be used to guide different types of minecarts into different tracks)

Minecart speeds :

• Default cart : 51 mph
• Iron cart : 66 mph (same as v1.2.4)
• Mechanical cart (from expert mode) : 102 mph

If the mecanist NPC is here, buy her 4 boosters (for 2 gold coins) and place them on the track, you can now get from 0 to maximum speed with the minecart.

Magic mirror/Ice mirror/Cell phone/Recall potion

It is cheap and teleports intantaneously the player who uses it to his spawn, but it only works in one way.

Hardmode Tier

The steampunker will be your best friend for these ones

Asphalt

Buy her a Blend-O-Matic to create a road where you can run fast on it, even without speed boots.
2 stone blocks + 1 gel = 1 asphalt block
In fact, asphalt multiplies your normal max speed, so if you're using speed buufs (sunflowers, potions, food), this max speed will be affected !

• Without anything : 53 mph
• With sunflowers : 64 mph
• With +45% speed buffs mentionned earlier : 77 mph
• 45% speed + sunflowers : 85 mph

After several testings, it seems that 85 mph is the maximum cap, and you may lose contact with the ground is you use slope with a speed superior to 80 mph.

Frozen slime block

Buy her a Solidifier to solidify 1 gel in a slime block and combine it with an ice block. There is no friction on those blocks, once you have enough speed, you can let go, but the goal here is to get the best speed, and you can do it without having to run :

• Use the hookshot's pulling speed : you can do it easily with hooking the middle of a door and then open the door after stabilizing.
  Remember that the bat hook has the best pulling speed from all the hooks.
  speed reachable : ~80 mph
• Use a cart. Being on frozen slime blocks in a cart won't keep your speed, you must exit the cart as soon as you are on slime blocks (you can exit it by going left/right when not on rails)
  Also, if you go on the wrong direction when on the slime blocks, you'll lose all your speed.
  speed reachable : ~100 mph (with the cart from expert mode)

Teleporter

The fastest way to travel in the game, and probably the most expensive. You need 2 teleporters on both sides, and connect them to wires and any kind of wire input to use them.

Mounts

Some mounts are very fast, like the unicorn one.
Using them will spare you the need of building anything, while still being faster than any wings/hoverboard.

the unicorn goes at 62 mph. In comparaison, the hoverboard's horizontal setting can only go to 52 mph horizontally.

Portal Gun

This little boy is dropped by the final boss at 100% chance, and when you hold this item in your hand, you can acquire momentum by falling at extreme speed, which can be converted into vertical, or horizontal speed.
However, you can't use the portal gun to teleport yourself to any far destination, A portal despawns when the player is approximately 1600 feet away (800 blocks)

• Vertical down : just fall with the gun in hand, you'll continuously accelerate until you reach 179 mph !!
• Vertical up : place a falling pit where you can reach 179 mph, and place the two portals on a ground, within the distance restriction : you'll end up rising at blazing speed. If you mount a flying pet and fly up as soon as you crossed the portal, you can keep the speed until you're done.
• Horizontal : falling pit, 179 mph, one portal on the ground, and one portal on the side of a wall next to frozen slime blocks. Warning : behavior with slopes might be flawed with this speed.

Game breaking Tier

HOIK glitch

It exploits the slopes mechanism for an optimal speed, and the cost is only basic, solid blocks !
You can use it at any moment in the game, but you can end up stuck in walls if you're not careful enough.
The maximum speed obtainable with this is :

• 2 blocks per frame (= 1/60 second), at horizontal speed.
  • which is 120 block per second.
• 3 blocks per frame, at vertical speed.
  • which makes an unbeatable 180 blocks per second.

For comparison, a 20 mph speed is approximately 16 blocs per second (measured with 131% of normal speed, no boots).
This would mean that a 179 mph speed (see portal gun, horizontal set-up) would reach around 143 blocks every second, which is faster, but it would require the player to defeat the final boss while the HOIK glitch can be done in very early game.

There is also a Magic Mirror which will teleport you from where you are to your set spawn point.

Nothing that works the other way round as far as I know though.