Schrader inner tube gets cut by Presta sized hole in rim I am new at bikes. I got an old bike with flat tires which I decided to put back to life. 

I went to Walmart and got the cheapest inner tubes (Goodyear) which turned out to be Schrader type. I had no idea about Schrader nor Presta valves, I was just getting new tubes.

When I started replacing the tubes, I saw that the valves of the old tubes were very different and my new valves didn't even fit the rim holes. I thought my bike was just too old and used some very old tube technology that nobody uses anymore.

I got rid of rim grommets and threw them away, and the new valves fit just fine. Now, however, the inner tube get cut around the valve all the time. What can be done to avoid cuts in this situation?
The cheapest solution is to try and remove the burrs or sharp edges that are cutting the tube. This can be done with a small file or a piece of emery cloth cut into a small strip and pushed through the hole. If the rim is double walled it will be difficult to get at the inner edges. The alternative is to purchase a set of schrader to presta adapter sleeves. While they are available on line a trip to your Local Bike Shop will verify that the drilled holes are large enough to accept the adapter. Then use Presta tubes and you should be trouble fee.