My back wheel isn't connected to the gears I have an old bike it says Via Montega on the body if that helps. I was riding it earlier when the pedaling stopped resulting in forward motion. 

I thought the chain might have come loose but it is definitely where it is supposed to be. The chain is attached and will shift gears but the back wheel doesn't turn. The wheel does spin freely if pushed. It seems like the gears and the wheel aren't connected but there is some sort of plastic guard between the gears and wheel so I can't see the problem. 

Is this something I can fix myself with limited to know bike repair knowledge or do I need to take it in? 

If so how expensive is it likely to be?
The bike is made by Univega and probably has a freewheel. 

If the pedaling motion still works, I'd guess your freewheel is shot (the thing that the gears are on in the back). You have to remove it with a freewheel remover and put on a new one. This requires a freewheel remover tool and a large wrench or vise, so you might want to take it to a shop and let them check it over and swap the freewheel out.