Bought Witcher 3 on the weekend
Do I need to launch Galaxy every time I want to play the game, or can I just run the game from the shortcut in the install directory? I don't like leaving Galaxy open because it makes my graphics card run at 60c for some reason.

I want you to do something for me. Right-click the shortcut and select "properties". Now, look at the target field and tell me; is Galaxy somewhere in the path? If no, then you can run the shortcut and the Galaxy client won't launch. If yes, then you'll need to rename the shortcut to remove the Galaxy references (or create a new one) in order for it to not launch Galaxy. This sounds like a strange issue, which I would recommend you report it to the Galaxy dev team. Instructions on how to do so can be found here: http://www.gog.com/forum/general/gog_galaxy_beta_2/post1

Shortcut points straight to the Galaxy client.
In the game manual it said I need to run it from that shortcut, so I wanted to ask first.
This is my first game on GOG.

No, the client is not necessary to run the game, contrary to what the manual tells you. You'll need to either create a new shortcut or modify the old one, so that the game will run without the Galaxy client. Your shortcut's target field should be "The Witcher 3 Wild Hunt\bin\x64\witcher3.exe" (modify appropriately to account for your drive's letter and installation folder).

Thanks for the help.