Just to build on Scott's tip.
Drilling rivets 101
1. Center punch the rivet with an auto punch right in the dimpled piont on the factory head. If there isn't one, get as close to center as possible.
2. Use a small drill to drill a pilot hole. As Scott describes, go slow. Pull it and look. If not centered, walk it carefully. Drill to the depth of the factory head. When your skills improve, you can go deeper to relieve stress so the rivet comes out easier.
3. Use a drill stop on the correct size bit for the hole. Leave only enough bit exposed to go through the factory head. This way, you can't drill the part and egg shape a hole. Drill the pilot hole.
4. Take an old bit the same size, insert the squared end and snap off the factory head.
5. Get a socket that just fits over the shop head with an extension. Use it to support the back side while you tap the rivet out with a punch.
An auto punch ground down go fit #3 rivet hole makes a useful one handed rivet driver. On the big #4s you will need a punch and small hammer. Don't beat them out. If they are stubborn, use the tiny pilot drill to carefully drill the shank.