Buy different pedals, or buy different shoes, or learn to love what you've got.
If you cut the bottoms of your shoes, they will fall apart very quickly. Also, they will probably feel like crap to walk in.

Look at MKS pedals, for examples of good pedals. The touring ones would probably work out OK, it's what I use and I ride in gym shoes. The lugs on the soles "catch" on the sides of the pedals so my feet don't slide around.
If you need dirt cheap, I've used aluminum pedals by Avenir (bought from Dick's) that were ~$10 and were just fine for a non-fancy bike. The Schwinn ones at Target and the Bell ones at Walmart are the exact same price, and they look exactly the same... they're probably all rolling off the same assembly line.