You will generally have a router (wifi and lan ports) and a gateway device (cable or DSL modem that connects to the wall). Increasingly these will be merged into one device. You generally want to disable logging in the router (merged or not), but there is some chance of logging in the gateway device (modem) itself. You can usually download manuals for the model of router and gateway device (modem) and peruse the settings. The router is generally what you'll want to toggle the setting in.
Beyond that you still have to be wary of OS level logging (and logging apps), your browser history (use chrome of firefox in private mode at a minimum) AND SAVED PASSWORDS AND BOOKMARKS. The possibilities vary by OS and browser.
Using cellular data with no wifi and the private mode of a browser is a good idea, along with having password protected accounts for all phones and computers set up as a 'normal thing' (with guest accounts where appropriate). Appropriate timeouts that lock screens should be set up. Also remember that any work provided internet, computer / laptop, or phone should be considered monitored unless you actually know what you're doing.
A burner phone strikes me as just asking to get caught, but YMMV.
