Replacement cassette for Schwinn Jaguar 7-speed? How would I figure out the correct replacement for a 7-speed Schwinn Jaguar cassette?

I'm newer to working on bikes (but lots of automotive experience).

Got this bike inexpensively. Chain keeps coming off. I gave the chainwheel a little adjustment with the hammer, looked like some of the teeth were out of alignment. Replacing the chain. The cassette looks pretty chewed up as well.

I see some $18 7-speed Shimano cassettes online. How would I know if I was getting the right one?
Clearly this post is in regards to a newer Schwinn rather than the originals from the '50s and '60s, as the latter were all equipped with Sturmey Archer internal-gear  hubs.  Searching for "Jaguar" on the Schwinn website yielded nothing so apparently Jaguars aren't manufactured anymore.  Getting a little more desperate, I consulted the Walmart website and found what I figure is a reasonable facsimile of a Jaguar and downloaded its manual, which suggests that what you're dealing with is in fact a freewheel, not a cassette.  The manual offers no insight into the brand, but you might look for a brand on the rear hub--the freewheel will have to match the hub, so matching brands there should get you what you need.  You might also look at the freewheel's tool interface (see Sheldon Brown link) to spot brand differences.