For years I have had great difficulties with drilling holes in stainless. I have burned many drill bits and used many interesting words over the years. I am sure there are experts out there that can help.
Speed, it's all about the speed. It's hard to go slow enough with hand tooks. On a drill press @2-300 rpm you would be amazed at how well it works. There are even new cutting oils that make a marked difference. It only takes a moment at to high an rpm to do in a drill bit.
You are not alone.
Stainless can be nasty to drill. Put on safety glasses. Clamp the steel well. Just use sharp ordinary high speed steel drill bits in a hand drill. Bring extra bits, as you will need them. If you want to step up, use a split-point cobalt steel bit or carbide bits if you are using a drill press. Start with a punch on your location, then use a drill half the size of your final whole, then your actual size drill.
Success is enhanced with good lubrication and keeping the drill cool. Slow down the spindle, to between 150 to 400 RPM and when you bring down the drill, drive it home forcefully.
The material tends to work-harden. Slow drill speed , and steady firm pressure on the drill to make it CUT, (never hesitate) so it will not scrape its way through. A spiral chip coming off that drill is better than making dust-sized particles.