$100 budget for new wheels, with new tires and tubes, installed? Ain't going to happen. Your best bet if you want to upgrade is to find a set of used wheels, either local Craigs List (post a WTB wheels in the bicycle for sale section), garage sale or a donor bike. Most of my wheels come off of donor bikes, and I have occasionally picked up wheelsets at garage sales (for $10 to $20 a set, often complete with tires). I am talking good, alloy road bike wheels, not crappy steel wheels or MTB wheels.
For brakes to work well you need alloy rims.
+1 You had 27 inch wheels on that bike. Tire choices are very limited. The good news is that there usually are a lot of used 27 inch wheels out there. I have not seen semi knobby tires in a 27 inch size, maybe someone else has. Assuming you have the bike shop do the work, half to 2/3 of your budget will go for just the tires and tubes (mounted).
Depending what you are trying to accomplish, you really would be better off with a nice LBS branded used rigid frame MTB. The 1970s Raleigh Grand Prix was an entry level bike. You can find nice MTBs around here at full market price for about $125. It is 5X the bike you currently have. And if you shop around, you can find them for $25 to $50. The tire choices for MTBs are just about limitless.
I run Kenda Kross Plus tires on my 1992 Trek 950 MTB, which is a nice tire for the type of riding you describe.