I have a Schwinn Tempo which has a 6 speed freewheel I'd like to replace my rear wheel because it isn't able to be fixed to be made true once again. The bike is an older Schwinn Tempo, the rear wheelset has a 6 speed freewheel. Or so that's what the guy at the shop told me. He only had wheels with casettes, and told me I wasn't able to use one of those. 

Question is, I'd like to get a new wheelset. Do I have to stick with the freewheel style? Or can I get a modern, casette type wheelset? If the latter, do I need to make any modifications? Adjustments?
A 6 speed bike has 126 mm rear spacing while more modern wheels will need 130 mm rear spacing (which you can fix by cold setting the frame). If you're getting a new front wheel too, you will need to do this to the fork as well (likely). If you go 7 speed and up, you can get a cassette wheel, but if your Tempo has indexed shifting, you'll need to use friction mode (or get new index shifters) and adjust the derailleur accordingly. 

The Schwinn Tempo has 700c wheels and there are decent 700c freewheels with 126 mm hub spacing made by Velo Orange under their Grand Cru line.