As of the latest update to Steam, you can now PERMANENTLY remove games from your Steam account!

Steps

1. Log into Steam Support
2. Select Games, Software, etc.,
3. Select the game from the search function that shows up
4. Select the option: I want to permanently remove this game from my account.
5. Select the option: Ok, remove the listed games from my account permanently.
6. Repeat as many times as you have games you'd like to get rid of.

You can browse over your entire library and choose which software you'd like to permanently get rid of. But note, once you have removed a game from your library, it's gone permanently, so if later down the road you'd like it back you'll have to purchase it again.

You'll also need to watch out for some other issues. The thread linked above has noted that if the game is part of a bundle (HumbleBundle, etc.), removing one game from a "package" may require you to remove all the games from that specific bundle.

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"