re: drill press
When you feel the bit start to go thru, let up on the pressure. Temper-mental little cuss, isn't it.
Another thing that will help is to have some kind of stop or holding device. Something as simple as a 1/2" bolt nutted in one of the slots in the table will help you keep your finger tips. A piece of 1/2" strap an inch or so wide with a hole in it bolted to the table and over the edge of your work piece also works. A while back I found a hold down clamp at a farm store. It looks like a pair of vice grips but has a lug on one jaw so it can be bolted down to your table. If all else fails, a good old fashion C-clamp works too.
Buying a drill press is like buying an air compressor, They're really handy but you damn near have to buy some other tool to go along with it.