Originally Posted by FBinNY

People often thing bike shops are just trying to sell more when they suggest (strongly) a new cassette or freewheel with a new chain. It's a sore point, but the simple fact is that a new chain won't run on a well worn sprocket, and an old chain won't run on a new sprocket.

If your original chain and sprocket are worn to where they're slipping, it's 100% sure that replacing either alone won't solve it, and will likely make it worse. You need both, and as someone else pointed out, you might need a new chainring or two, but these are more forgiving, so you might skate on that.

Most shops, only charge nominal labor to install a chain and cassette if you buy both from them (when I was in retain, we'd do the install free if both were bought new). Or you might search locally for a bike co-op where you can do it yourself, using their tools, learning in the process.

If you had listed your city of residence in your profile, someone could likely have steered you to a decent shop or co-op in your area.

I assume this is an older bike and understand that you don't want to spend any more than you absolutely have to. Look for a "blue collar" bike shop, where they specialize in more basic family family and entry level bikes and repairs, than a "pro shop" which caters to a more monied crowd.