What's the best way to update Homebrew when upgrading macOS? I'm just about to finally update my MacBook Pro to macOS Mojave, hoping that the issues have all been resolved. But when I got this new machine I changed from using MacPorts to using Homebrew.

When I used MacPorts, I had a (semi-successful) recipe for handling an OS upgrade: save a list of the ports, remove the MacPorts installation, install the new OS, install the new developer tools, install MacPorts, and then slog through the saved port list and update (since inevitably the set of ports would have changed).

What's the corresponding recipe for Homebrew? I have searched for "brew update Mojave" and found nothing.
Homebrew manages all updating/upgrading by itself. Run brew update &amp;&amp; brew upgrade every once in a while (and you can do it after upgrading macOS). brew update will update the list of available formulae, and brew upgrade will upgrade any outdated packages.
If you like, everything can be scripted as well:

Run the macOS installer: startosinstall
Run all macOS updates for Xcode, etc..: softwareupdate -ai
Update homebrew itself and the package lists: brew update
Upgrade all software installed with homebrew: brew upgrade
Remove old versions of installed software: brew cleanup

You don't need to remove/uninstall anything before upgrading macOS. Just download the macOS installer from the App Store (or from System Preferences), and follow the instructions to install the new OS like normal.