I'd like to have a modded and a non-modded version of Skyrim, and be able to switch between them (say if my wife wants to play a modded game and I want to play vanilla).

What I did so far is just copied & pasted the Skyrim folder onto my desktop. Next step would be to disable mods, but I've noticed that disabled mods on my copy also get disabled on my original.

What would I need to do to make it possible to have different setups?

Look for the Mod Organizer, here is one thread on it

http://www.gamesas.com/topic/1485255-mod-organizer-good-to-use/

And, see this

http://www.gamesas.com/topic/1264910-wipz-mimi-multiple-install-manager-interface/