Using a Drill Press
It is best to have a drill press if you are doing a lot of drilling. It provides a stable drilling action and the pressure on the bit can be controlled. It is important to ensure the bit is running true without wobble. The drill press should have instructions to help correct any untrue running of the chuck. Make sure the drill bit is secured firmly. Core drill bits are easier to keep true, as they normally have a threaded fixing.
With a drill press, you can drill continually until the hole is completed, or until a white paste or dust begins to appear. This indicates the drilling is being done dry and will in a few moments heat up the glass too much. When the white paste appears, back out of the bottom of the hole a little to allow water to flush the glass out. Then continue.
Keep a firm grip on the glass being drilled. Maintain its position, especially if you are intending to back off intermittently to allow water to the bottom of the hole. This enables you to get back into the hole without scratches.
If possible, submerge the piece. But if that isn’t possible, just squeeze a little puddle of water on the surface and watch it swirl around. You can see if it is pulling ground glass out of the hole by watching the circulation. Placing a plasticine or clay dam around the drill area will keep the water confined.
Don't push down any harder than you comfortably can with the tips of your fingers. Keep it steady. Listen for the sound of diamond grinding glass
If the core gets stuck in the bit, knock it out with some stiff wire or a nail. Always remove each core right after drilling. They are very difficult to remove if there is more than one in there.
Every diameter drill bit has an optimum drill speed. The smaller they are the faster the speed required. Based on what the manufacturers recommend, a Dremel running at top speed is way too fast. When using a Dremel for drilling glass, slow it down with the speed control. Drill presses do tend to be on the slow side for glass drilling so it takes a bit longer, but there are big advantages in other respects.
For other tips on glass drilling see:
Keeping things wet
Using a drill press
Drilling with a Flushing Head
Drilling holes with copper tube and grit
Drilling glass without a drill press
Drilling speeds for diamond bits in glass