I’m cleaning up my hard drives, getting ready for a new PC build, and I found that I’ve got my Steam library littered everywhere: C:\SteamLibrary, C:\Games\SteamApps, C:\Program Files\Steam, D:\SteamLibrary, etc., etc.
I started to sweat at the thought of manually backing up saved games and control inputs, uninstalling, and then going through the pain of redownloading everything again, but I found this tip:
tl;dr, copy your whole game folder to the new location, “delete local content” in steam, then “install game” to the new location. Steam is smart enough to discover existing files and, voila - your games have moved.
[edit: moving some of the bigger games, like Elite Dangerous, Batman Arkham Origins, does take some time, but a lot less than redownloading and reinstalling!]