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I'm going to update a multisite WordPress installation from WordPress 5.3.18, and plan on updating it to WordPress 6.6, but I want to do this somewhat gradually with a few days in between. So I'll have a bit better control over the work needed to be done to our themes and plugins, and I'll be able to identify which versions introduce which problems, et cetera.
However, going through every major version between 5.3 and 6.6 seems a bit excessive. Is there like 3-5 recommended versions that would good to use as stepping stones for this process?

You’re probably best off just updating everything all at once on a staging server and skipping the different stages, you’re just creating more work for yourself. Why test everything multiple times when you only need to do it once? If your PHP Version isn’t up to date, you may as well do that at the same time too – you’ll need to test backend and frontend thoroughly anyway.
There’s no need to update each version – just hit Update. Take a backup first though, and do it on a staging site if you’re concerned about any old plugins/theme.