If you pop on a new chain, and the cassette cogs are worn out, you will get "skipping" when you pedal in the most worn gears. My guess, based on your info, is that you need a new chain and cassette. The front chainrings are probably ok.
Chains wear faster than cassettes, so if you replace the chain before it's completely shot, you can keep the same cassette at least until you replace the chain a 2nd time. Depends on how well lubed and clean your drivetrain is.