Before January 19th 2017, there was no built in method to move games between libraries

Here is what I did though:

1. Browsed to the steamapps folder that the game is currently installed into
2. Copied the game folder for the game I wanted to move into the other Steam library (it's important you copy it; don't move it)
3. Deleted the game within Steam (delete local content)
4. Reinstalled the game, selecting the other game library

This finished "downloading" to the new location instantly, since the files were already present in the new location.

Important note on the folder structure of the destination library

It is important to note that as of trying this process on 2/11/2016 you cannot simply paste the folder into the destination "steamapps" folder but instead to a "common" folder within the "steamapps" folder.

For example if you created a Steam Content Library at "D:\Games\Steam" you will want to paste your game folder into "D:\Games\Steam\steamapps\common" resulting in something like "D:\Games\Steam\steamapps\common\Grand Theft Auto V"

Steam has probably made a library folder on the drive that you installed the Steam client on. If you don't have any games on that drive and don't intend to ever install any on it, go to Steam>Settings>Downloads>Steam Library Folders, select the library folder of the drive you don't want to install any games whatsoever to, and click "Remove".

Furthermore, you can use the same method (except you click "Add" and not "Remove") to add another Steam library folder.