Bike Noob: Need Help with Drive Train Rebuild

I have what I believe to be a 1991 Specialized Rockhopper that I bought at a local non-profit. The chainrings are rusty and missing a few teeth, and the rear cassette is very worn. I would like to remove/replace the cassette, chainrings, and chain. It currently has a Biopace chainring, everthing looks original except the shop replaced some of the rear sprockets individually. I found some info from another site http://www.anthonyalsberg.com/bike_spec/92_93/rock_fs.html:
Drivetrain:
Derailleurs -- Shimano STX Special Edition, Grip Shift SRT 500 R shifters
Chain -- Shimano Hyperglide
Crankset -- Shimano STX Special Edition, Shimano Hyperdrive C rings, 175 mm crankarms
Bottom bracket -- Shimano Cartridge
Freewheel -- Shimano 7-speed cassette

Where do I start? Count all teeth on rear cassette, count all chainring teeth, and order matching HG parts? 7 speed Shimano hyperglide chain?

I would go to a bike dedicated forum, but that is no fun . Thanks!

 

Does it shift through all the gears?

I'd start with general cleaning and then assess what needs to replaced. No need to replacing anything if it works and just needs to be cleaned. Usually replacing the chain and cassette or freewheel is in order for a worn out drive train.