With GitHub how do I push all branches when adding an existing repo?

Note: git push --all won't push your tags, only your branches.

git push --all
git push --tags

would really push everything. See also "Set up git to pull and push all branches".
Don't forget the --dry-run option to make some test before actually pushing everything.
See also GitHub help "Working with remotes" to set your origin to your GitHub remote repo.

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As mentioned in "How to make “git push” include tags within a branch?", git 1.8.3+ (May 2013) introduced:

git push --follow-tags

This won't push all the tags, but only the ones accessible from the branch(es) HEAD(s) you are pushing.

That can help keeping that operation (pushing commits and tags) done with one command instead of two.

Git 2.4.1+ (Q2 2015) will introduce the option push.followTags.

The accepted answer isn't quite right, or at least it didn't work for me. I needed to specify the remote repo as well, eg:

git push origin --all