The freewheel mechanism (the part that whirs when a bike coasts) needs to be lubricated. Unfortunately, this isn't something easy to do.

It could be that the bike has sat unused so long that the grease dried and got sticky. Simply riding the bike might loosen this up.

Your other option is to get a new freewheel. This might lead to a cascade of new parts: chain, maybe chainrings, you said the rear wheel is wobbly so a new rear wheel, new tires, of wow that's a lot of patches on those tubes, shift cables cause the old ones are rusted, cable housing, while you're at might as well get new shifters, the old grips fell apart getting them off to change the shifters, and oh that saddle looks crummy, oh drat the seat post is stuck so cut it out, ... And quite soon you've paid for a new bike but still have the crummy old bike.