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In practice, I believe the private key should contain the public key (or at least sufficient data to recover it): https://superuser.com/questions/814409/gnupg-opengpg-recovering-public-key-from-private-key#814421
I believe you only need your private key to sign files so, technically you only need to back up the private key, but you should test this to be sure it fits your use case.
Depending on how you're backing things up, and what your security goals are, remember that backing up a private key may involve putting that private key on somebody else's computer - i.e. if you use a remote git repo, or cloud backup service, or even send the key to your own (different) machine over an insecure network. Make sure that you've got a way of securely backing up your private key, otherwise you may undermine the whole cryptography thing anyways :).
As always, you should test by backing up your key(s) and then testing that you can actually restore them and successfully sign a file. Backups are only as good as the last time you tested restoring from them.