GIT - rename a local and remote branch

First checkout to your old branch:

git checkout old_branch

To be sure that you have a branch up to date:

git pull origin old_branch

Rename local branch:

git branch -m old_branch new_branch

Set upstream to track your new named branch:

git push -u origin new_branch

Remove old remote branch:

git push origin :old_branch

More details

-m option is clearly explained in GIT documentation:

With a -m or -M option, <oldbranch> will be renamed to <newbranch>. If <oldbranch> had a corresponding reflog, it is renamed to match <newbranch>, and a reflog entry is created to remember the branch renaming. If <newbranch> exists, -M must be used to force the rename to happen.

You can also use -c option to copy your branch instead of renaming.