How do I make it so logging out from youtube doesn't log me out of gmail? (I don't want the two connected. And talking same browser window)

You can clear the cookies for YouTube and YouTube only.

You can remove cookies like this in Chromium-based browsers by clicking on the lock in the address bar, cookies then click remove for every YouTube domain.

This will make YouTube lose its state, such as the authuser currently using YouTube.

But please note that, since all Google apps are connected, you can be logged in again without a password by clicking on the Sign In button (this login is even automatic on some supported browsers) using the defaut authuser (authuser=0) whenever you visit the site again, so there's really no point of logging out of one service.