PCB drilling - drill bit material selection
Whatever size drill press you use, from a small Dremel press mount to a floor-standing shop drill press, you want to make sure that the chuck does not wobble. I originally used a cheap benchtop drill press, but the chuck wobbles about 1/64" (.4 mm), which is inadequate for most PCB drilling.
You should look for carbide bits instead of high speed steel (HSS). Tungsten carbide is great for PCB, but they are brittle. They won't become dull as quickly when drilling FR4 PCB material. Having a good press that doesn't wobble helps a lot in avoiding broken bits.
For holes < 1mm, sometimes even the Dremel can be too unsteady, and break bits. You might want to look for a "jewelers drill press" which is designed for small precision work.
One example is the Microlux mini drill press that Micro Mark sells, though you can find similar ones for sale on eBay.
I have exactly same PROXXON bench drill at home. Most of the times I drill with 0.65mm HAM 380 (cost about 5 EUR). It feels like it can stay sharp forever. In my experience it was like new with over 500 holes. I have broke few drill bits as well, but it was just my fault. Also I have found that it's quite uncomfortable to use collet set, so you may want to consider buying Proxxon Drill Chuck (28122) as well.