I drill through glass all the time, and here's the setup that suits me best, and is the most efficient (it works great for stones as well):
I use a regular Dremel (or comparable brand) high-speed hand-held drill. If you don't already have one, you can get one at a hardware store. Don't get a battery-operated one because it won't go fast enough. Get one with a cord. You want the model that goes up to 30,000 rpm. The big warehouse chain stores often sell Dremel kits with lots of attachments, but you won't need that for this. Metalsmiths who have jewelers' flex shafts can use them instead, but I didn't want my flex shaft to be sprayed with water, so I invested in the dremel drill which I use solely for drilling holes.
Once you have your high-speed drill, you need to obtain a chuck that will hold tiny drill bits. Dremel now sells a universal chuck that will accept any size drill bit--it comes in one of those little Dremel accessory packages that you can find at the hardware store.
The next piece of equipment is optional: a drill press. Basically, it's just a stand with a special clamp that holds the the body of the drill in place vertically, and a lever handle to move the whole thing up and down. You can drill by hand, but the drill press saves you so much stress because it keeps the drill perfectly in place as you're boring through the glass or stone.
The last component that you would need to get is a diamond drill bit. I use 1.2 to 1.5 mm bits, and I buy at least three at a time because they are breakable. But they work the best and the fastest. You can probably get them from a local jewelry supply store or online. If you need to make bigger holes, I would still suggest starting with the smaller bit, and then enlarging the hole with a bigger bit. This approach allows you to "fine tune" the location of the hole. Plus, it puts less pressure on the stone that way, and you're less likely to break the stone.
Here is what I believe is the safest, best way of doing this:
1. After I've set up the drill and marked a tiny dot on the stone where I want to hole to go, I put the glass or stone piece in a small plastic tray (like something from a frozen dinner) and pour enough warm water into the tray just to to cover the piece. (If you're drilling glass, make the dot with a permanent marker, then rub a bit of clear solid deodorant over the dot to keep the water from washing it off).
If you are not used to drilling, you might want to put a small extra sheet of plastic under the stone or glass piece at first, to keep you from drilling a hole in the tray! I actually use a plastic phone jack wall plate because that has a couple of small holes in it already, so when the drill goes all the way through the stone, it will next hit the hole in the plastic plate and I can stop the drill before puncturing the tray underneath.
2. I put on my protective glasses and a face mask (ALWAYS wear these), and then put the tray at the base of the drill press. I get a comfortable, firm grip on the stone that keeps my fingers out of the way of the drill. While holding firmly on to the stone, I bring the lever down until the diamond drill bit just touches the stone. The drill is not running now; I'm just trying to get the stone in exactly the right position. I always check also by looking at the drill bit from the side, as well as from the front.
3. When I'm satisfied that the drill and the stone are exactly aligned,and then turn on the drill to a medium speed (for stone) or a medium-high speed (for glass). (The general rule of thumb is: the harder the material you're drilling, the faster the drill speed. Diamond bits usually work best at medium to high speeds. You'll soon get a sense of what works best for you).
4. I slowly bring the running drill bit down to the stone and apply very gentle pressure for maybe about five seconds. Then I lift the drill bit up slightly so that the surrounding water can rinse off the diamond coating and cool the bit. A second or two later, I bear gently down again, then lift up, then press gently but firmly down, slowly and gradually working my way through the stone. Depending on the hardness of the stone, it may take you about three minutes to get all the way through the stone, glass even longer. You Have To Be Patient. You might want to consider investing in earplugs or ear protectors because it's a very noisy process.
If you find that your hand is cramping or the stone is slipping out of your hand, lift the drill completely up and turn it off. It will be easy to correctly reposition the stone; all you have to do is bring the drill bit down (turned off), and you'll feel it slide across the smooth surface of the stone and catch when it enters the drilled hole. You can then turn on the drill and resume working. Again, if you apply too much pressure and try to go too fast, you will risk breaking the piece. Remember that you have to have water filling that drill hole at all times, or else the diamonds will spall off of the bit and render it useless.
If you're drilling stone, there's no guarantee that it won't break even if you do this method, because some stones may contain internal fracture lines that will break the stone apart with the pressure of a drill. So you should decide at the start whether you want to risk the stone or not. I'd suggest practicing on cheaper, throwaway stones at first. And, of course, stay away from drilling in "cracked" areas of the stone.
You can also do this without water, using special jeweler's lubricant instead, but water is the method that works best for me. I hope it works well for you, too.