Vinyl and Linoleum Flooring - Floor replacement

I'm on a budget and always attempt to try things myself rather than hire it done. I decided my 10 year old vinyl needed to go, so I pulled it up. I'm left with the backing that is glued down to remove. I can see the original vinyl floor from 1965 under it. Under it is a plywood subfloor nailed to the concrete slab. Should I try to remove the old vinyl or the complete subfloor? I haven't decided if I want to put down ceramic tile or vinyl tiles. Any suggestions would be appreciated.

You will want to scrape it all off as much as possible. If you are going back with vinyl, some advocate layering luan over the glue residue once you have scraped it so any aberrations won't telegraph through your new vinyl. If you choose ceramic , get it down as good as possible, put down thinset and concrete backer underlayment, then you can apply your thinset and tile, then grout. Much sturdier floor with the ceramic, but that is an opinion.