Road bike wheels...?

OK OK so dont give me hell about owning this bike. I have a Schwinn Prelude from Target. I am now seeing that the wheels/ rims are a problem. They are constantly popping the inner tube. I can't afford (right now) the bike I really want (a 650 Felt), so I was thinking about buying new wheels. I need to find quality wheel set for cheep, like 100 bucks or less. I would like to buy new wheels so I can ride for now and save for a new bike later.
Or if I cant find wheels what about the rim tape? Can I double or triple the amount of tape on the rim to protect against popping the innertube. All I did today was take my bike from the trainer inside to the garage to inflate the tires, left home and came back and the tires were blown out.

Or if there is something else I can do please suggest.

yea its kinda like blown out.. there are holes in the rim where the spokes are... the spokes are tight.. checked thoes.. when I inflate the innertube the pops or blowes are big holes where the holes in the rim are it looks like the inner tube is being sucked into the hole and blowing...

Steel rims are the cheapest and heaviest available, aluminum is better, and composite it the toughest/most expensive.

Are your spokes digging into your tubes?
Either new rim strips or 3 to 4 layers of electrical tape will care of that.

Are the sides of your tube getting pinch-cut by the rim?

To keep your tube from getting pinched between the rim and
a curb, you need enough air pressure to keep the rim off the road. If you are heavy, you can usually get away with about 10% higher air pressure than the maximum on the tire sidewall. The ride will be rougher, but the rolling resisitance will be lower.

If you are getting flats from road hazards,then those puncture proof plastic strips you put between the tire tread and tube will do the job.

The stock wheels that came with the bike should be OK for casual riding.
First thing to do is check the tube. If the puncture looks like a snake bite, it's a pinch flat, probably caused by under-inflated tires. Keep your tires inflated to the pressure recomended on the sidewall. If the puncture is a single hole, it's caused by a sharp object. When you change a tube, carefully run your finger along the inside of the tire to check for sharp objects that might still be imbedded in the tire. A single layer of rim tape should be sufficient, but it won't hurt to double it up. When you changed the tire last, did you use tire levers to get the tire back on the rim? If so, you may have punctured the tube. Always resist the temptation to use tire levers to get the tire back on, always use your hands to get that last bit of tire on the rim.

Buy some Neuvation M28 Aeros or Mavic Aksiums and you'll be set. Both are fairly decent wheels at relatively low prices. The Neuvations are actually on par with the more expensive Mavic Ksyrium Elites, but for much less. Neuvation has a sale on wheels on their website neuvationcycling.com right now. Check them out.

I bought the mavic silver wheels as a set from performancebike.com @ $200. Had them now for 3 years with no trouble.

A lot of the info provided here is good, but one thing to look for is an embedded piece of glass or metal in the tire itself. It is hard to find unless you:
a - remove the tire from the rim an literally turn it inside-out (forcing the material to pop up on the inside of the tire) or
b - carefully run your finger inside the tire feeling for any shaprt/p[rotruding objects.

The worst offenders are tiny glass shards, small sharp chunks of rock/gravel (depending on the part of country you are in) and small staples or nails.

Check to see where the puncture is, to do this remove the tube, (remembering it's relationship to the rim) inflate the tube and locate the leak. If the leak is where it would come in contact with the tire carefully inspect the tire for something in the casing (thorns, glass, etc.). If the puncture is facing the rim look in the area for the cause. The rim tape might be the reason as these bikes come with a poor quality tape.

first what is actually doing the poping here pointy spoke or metal shard. Take your rim tape off and run your hand round the rim (i know it sounds stupid but check the tyre as well with ur hand) can you feel the specific spoke or area that is popping the inner tube, if you can get a file to round it off etc or improvise you say you don't need these wheels well who cares what you do to them so long as your inertube stay inflated.

other possibilities are you need to tension you spokes to stop them being so loose, if there not sharp its possible your getting pinch punctures with them being to loose. get a truing key if you can and tighten them (if you do that read up on it cause its all back to front) .

hmmm i just re read your post blown out as it actually exploded like massive hole rather than puncture?? in that case run your hand round the rim again and check the sides, the bit where your tyre hooks under the metal rim check for any deviations or places where the tyre wouldn't get a gd lock. Those spots are called flat spots and are cause from a big thump (the more expensive the wheel the bigger the thump it can take normally) theres nothing you can do about them other than improvise and try and return it to the original shape.

still reading? well if its none of those try your actual tyre does it have a gash in the side/ warn through these spots dont hold pressure very well and can cause the tyre to slip off the rim the inner tube then exploding like a ballon.

lastly take it to a gd bike shop when i worked at a bike shop people brought in stuff all the time for us to look at (even un bike related things) and people dont charge you for looking and diagnosing problems like you've got.