Easiest fix for this transmission problem? My bike is an adult mountain bike with different gear shifts and all. 

Suddenly my rear tire stopped going but the chain and everything still works - the gear and chain still spins.

What is the easiest way to fix the rear chain/cog if it's not turning the rear wheel?
If you mean that that the cassette can turn both forward and backwards without engaging the rear wheel hub; the freewheel or freehub is broken.

Whether the freehub/wheel can be repaired, or whether the hub or possibly the whole wheel would need to be replaced depends on a number of things. Suggest you the bike to bike repair shop.