Wheelset Sizing
 

If I want to replace my entire rear wheelset, do the specs have to be exactly the same as what I am taking off? The old wheelset is 26x1.50HE VP20, but finding anything that seems similar is proving to be difficult. Is there another size I can use?

That's a pretty generic rim on older (at least) mountain bikes.
It gives a lot of latitude in choosing different tire widths.
IF you want something quite similar, look for a rim that's about 20MM wide.

The only critical dimension is that it's a standard mtn 25x1.xx rim (559) rim. That's critical because affects brake height and any change can take the rim beyond the reach of your brake shoes. Rim width is the second factor, and here there's more latitude & you might want to choose the rim width according to the tire width you plan to use. Here's a link to a chart that will give rough guidelines.

Note that rim width can also affect the brakes, since braking is best if the side of the rim is the right distance from the canti post. However there are ways to compensate, so fee free to change the rim width to best suit your needs, and you'll compensate the brake later.

That was definitely helpful, thanks. I think I'm going to try and stick with what I have, I'm upgrading the freewheel sprockets first.

Do you have one of these too?

Wheelset refers to a pair of wheels.

Haha, whoops! Good catch.

IF you are replacing the wheel, why not go to a Free Hub and cassette?
It's a win-win.
Cassettes come in a wider variety of cog combinations.
Free Hubs aren't prone to bending/breaking axles.

http://sheldonbrown.com/free-k7.html

That is what I'm trying to get to, eventually. I'm mostly trying to get it on the road right now, and without replacing the actual wheels/hubs I can get away with just replacing the freewheel sprockets. I'm not sure if I'm going to keep upgrading this bike, or just find another.

The hubs aren't compatible, so if you get a different FW now, you won't be able to use it when you upgrade to a free hub wheel.
It'd save money in the long run to upgrade both now.
You can find rear wheels for about $45 online.

What's your dropout spacing?

I have no idea what my drop-out spacing is. Sorry! But I know they aren't compatible, my wheels aren't in terrible shape as it stands, but the FW is rusted out and so is the chain.

A new freewheel can cost $10 or $15. A chain similar. If this is all you need then get it. Wheels with freehubs and cassettes have advantages, but if you have wheels in generally good condition already then replace what needs to be replaced and go ride.