I just want to make sure I do it correctly and safely... I'm getting a new laptop very soon (like, end of next month), so I still need to know how to uninstall it properly from this family computer, because sometimes, programs like to leave little bits behind..

Wait until you have your new laptop - install Origin on your new one and download UC from Origin on your new laptop first - then uninstall it the normal way on the family pc, by going to programs and features in your control panel. You might want to uninstall Origin from it as well, if nobody else uses it. (Upgraded very recently myself).

Uninstalling through Origin is the best option indeed, as that'll prevent problems with the DRM, if there is any. You don't need to wait until you have your new laptop if you don't want to, though; just remember to back up My Docs/EA Games before uninstalling, because Origin will delete your sims user data when you uninstall.

You might also want to uninstall Origin through the control panel if no-one else uses that computer to play Origin games.