Sorry to confound things, but depending on the age of your bike, with 7 speeds in the rear, you may have a cassette and freehub rather than a freewheel. Same problem, but the solution would be different. Take another look back there, focusing on the outside end where smallest cog/sprocket is. Does it look like The Golden Boy's pic above, with a recess around the central opening? Or is it more flat:

like this? If it's the latter, you have a cassette, a set of sprockets that slides over the freehub. You'd have to remove the cassette (which requires a couple bike-specific tools) to reach the ratcheting mechanism of the freehub to lubricate it.

Do you have a bike co-op / "bike kitchen" where you live? If so, there's probably a day you could go in and use their tools, and sponge off their expertise, to determine what you have and how to lubricate it.