What everybody else said, except for that part.
You need to make sure that the drill is cutting all the time and not skipping/rubbing over the surface without cutting, because that's when too much heat is generated and the drill is then probably toast. If it's cutting right you should see continuous spirals of steel coming out from the drill flutes. Be careful just as the drill starts to break through at the last moment, it will start to cut much deeper because the point is now in fresh air and the pressure on the outer parts of the cutting edges increases rapidly. At this stage the drill may bite off more than it can chew in which case the torque from the drilling machine might break it. You should hear the motor slow as the cutting load from the bit increases, and you may feel the torque increase. Time to ease off the pressure and perhaps slow the speed if you have a variable speed trigger.
You need a good quality drill (there are some real rubbish Chinese drills out there) but as Biker says, HSS is fine if you use the right techniques. If I'm drilling stainless by hand (a powered hand drill I mean) I use something like a squeezy bottle full of water for coolant. If you're inexperienced then for safety's sake it's best to get a friend to squirt it while you drill. You don't need gallons of water but the drill and work need to stay wet. Steam will probably rise from the drill, that's all right. If you can arrange a small flow of water from a little hose or something so much the better. Dry the drill bit afterwards or it will quickly be damaged by rust.
You can get drills ground to a different geometry for drilling stainless but it isn't worth it for what we're doing here.