Create the repo on gitlab. (log in, create it in either your space or snac's space, and then copy the URL for cloning via ssh (git@gitlab.iath...)
On your already cloned github repo (working directory), add gitlab as a remote:
git remote add gitlab git@gitlab.iath... (copied from earlier)
Push every local branch to gitlab
git push gitlab --all
You might have to change your upstream on a branch if you want to track gitlab instead of github:
git branch --set-upstream-to=gitlab/master (or whatever other gitlab branch you want to track) This sets the current checked-out branch's upstream to whatever is passed.
Now, when you push/pull/fetch, you can choose what you want to do:
git push origin master pushes to remote origin, which is github
git push gitlab master pushes to remote gitlab
git fetch --all fetches both gitlab and github
git fetch gitlab would just fetch gitlab (similarly for origin)
git remote remove origin would remove github as a remote.