You may want to pick up a center drill or drill/countersink like pictured below.


You can center punch the stud and use the center drill to start the hole. I use them all the time and they help keep the drill from wandering.

A cobalt drill bit will hold up to the stainless better. You can either resharpen the bits by hand or use a drill doctor. I've been using a drill doctor for several years and it works well for re-sharpening bits. I also use it on solid carbide drills to.

Good luck with your project.