I run Lighthouse and WAVE as a “Hey, let’s see what I have ahead of me” kind of thing. A baseline of sorts. Then I go into manual testing
I strongly disagree with running automated tests first if you have the means to do manual tests. People get too hung up on automated tests revealing only N percent of issues; the reality is that they end up encouraging people to spend their time fixing those issues instead of more critical errors.
Less than a minute with the “inspector” tool in most browser DevTools (not even the Accessibility Inspector!) will quickly reveal poor use of semantic HTML, poorly-written alt-text, site titles that shouldn’t be headings, non-descriptive names, and conflation of semantics with presentation. Fixing these is probably a better use of your time than hunting down every duplicate link name in a page that an automated checker finds.
I prefer running an automated checker after catching low-hanging fruit manually.