I just use a Cobalt drill and it works fine. It needs to be sharp, though. Unless it has to take a big load I usually go one size larger than the table calls for.
I agree with taper taps.
When drilling stainless, the trick is to not dwell. Reduce your cutting speed to about 80sfm. Start drilling and keep up your feed. You can go about 1-1/2 the drill diameter before clearing the chips. When you resume drilling maintain your feed. If the bottom work hardens, increase your feed pressure to punch through and then try to keep up with the feed. You are doing this to avoid work hardening the hole so you can come back and tap it.