First: I really don't know how to put together a network, so bear that in mind with the question.
Not a big problem, once you describe your network equipments, I'm sure you will find all help to do the right setup.
Question: (a) If I buy a printer with an ethernet connection, will it print off my regular router, and what would I have to do to make it work with more than one computer--one Windows, one Linux.
Do you mean a router with an incorporated switch ? Yes, consider it a transparent device between computers and printers.
(Unless you need to print remotely over the internet, that's whole different story).
(b) If instead (or in addition) I bought a printer with a WI-FI input, would it work with a WI-Fi router, and what would I
have to do to get it set up and working? (Same two computers.)
You will need to configure the printer wifi connection, depending on the printer may be done using the printer panel or a built-in http server, or (worst case) a windows software, other than this, the same rules as for a wired network should be applied.
thanx for any advice along these lines. --doug
AS