As already stated the rear cassette is worn out, check the teeth on it against a new one.
You might need a special tool to get the cassette off, at least a chain whip, probably the bike shops don't have it.
If you replace the cassette you will have to replace the chain as the old one will slip on a new cassette, the front chainwheel looks to be OK.
Suggest go to a more up market bike shop who will have the tools and knowledge to do this simple job.
I know all this because when I retired I pursued my bike riding hobby by importing mountain bikes into Singapore, I also owned five retail mountain bike shops there as well