The quick, easy and complete way:

1. Clone the repo you want to move or skip this step if you already have one set up:

    git clone git@oldreposerver:repo/name <insert dir>
    cd <insert dir>
   

2. Make sure your repo has all the latest news from its current origin:

    git fetch --all 
   

3. Add the new origin as a temporary remote whilst keeping the old one for now:

    git remote add newrepo ssh://git@newreposerver:newrepo/newname
   

4. Push all refs from origin to their equivalent counterpart on newrepo:

    git push newrepo '+refs/remotes/origin/*:refs/heads/*'
   

5. Switch origin around so your new repository becomes default:

    git remote rename origin oldrepo
    git remote rename newrepo origin
   

6. Or get rid of the old origin altogether if you’re positive you won’t be needing it anymore:

    git remote remove origin
    git remote rename newrepo origin