So, one of the most annoying problems with the game for me so far is the massively slow move speed. In the Sister's chambers you can move as fast as in the other chambers by holding shift, but that's all.

So I was watching this: http://www.youtube.com/watch?v=2Awni-pv87U

He seems to move at least twice as fast as I can in the mine. Is there some button I'm meant to hold down to move faster?

The real problem is the chambers' size. Your walking speed is quite big, I would say. It's about 10-15 feet per second. But the chambers were made huge to coempensate exactly for that. Then we made the speed bigger, but after that we constructed even bigger levels and so on .

First, you don't need to use "Shift" in usual chambers (it slows you down in all chambers except those where Sisters reside). Second, the speed of your movement increases if you have much Azure in your hearts.

As an alternative, you can tweak the variables used by game in Properties/player.xml . Use Total Commander plugin to get access to the game's resources inside Properties.vfs. Then simply create subfolder /Properties inside /data and put any changed files there.

Was it something changed in the versions? I'm using the Steam version, the guy in the video isn't, and he has no Azure in his hearts. I have to use about a 30 colour haste to get the same speed. Maybe something that can be patched in the Steam version (and the option to skip the tutorial too!)?

No, there weren't any changes. Percieved speed depends on many things, such as your resolution/screen size and the size of the chamber as compared with the movement speed. The same very speed feels quite fast in the first garden, but slow in "Fireflies", and even slower in "Lemna". If you are in a hurry, there is really no choice but to fiddle with default numbers set in game's resources.
Download the file here: http://www.megaupload.com/?d=DUQA9S73 . Put it inside /Void/data/Properties/ (I suppose, in Steam your game will be inside /Steam/Steamapps/YOURNAME/Void/)

It's in "Steam/steamapps/common/the void" here, just like most third party games purchased on steam. Even most games that fully use the fancy steam cloud things lie in "steamapps/common"