I would suggest:

doc8 README.rst
doc8 docs/source/
make -C docs html SPHINXOPTS="-W -E"
make -C docs linkcheck

I think it would be great to be able to run tests locally with Tox (and make GitHub Actions run Tox in CI). But that could be a separate issue. See for example osbrain's Tox configuration and how Travis runs just Tox.