If you want to rename a branch while pointed to any branch, do:
git branch -m <oldname> <newname>
If you want to rename the current branch, you can do:
git branch -m <newname>
If you want to push the local branch and reset the upstream branch:
git push origin -u <newname>
And finally if you want to Delete the remote branch:
git push origin --delete <oldname>
A way to remember this is -m is for "move" (or mv), which is how you rename files. Adding an alias could also help. To do so, run the following:
git config --global alias.rename 'branch -m'
If you are on Windows or another case-insensitive filesystem, and there are only capitalization changes in the name, you need to use -M, otherwise, git will throw branch already exists error:
git branch -M <newname>