For that style machine, a laminate trimmer or small router (like the Bosch Colt) would be a good choice. Trying to hang a full sized router on that frame might be asking a bit much, but it's hard to say from the pictures.
You definitely don't want a hand drill on there. They're not designed to take the sideways loads that milling will put on them.
It looks like several builders have converted from belt drive to lead screws. I don't know their reasoning, but you should consider how you want to drive this machine. Stick with the original plan, or go with screws?
I've used the Xylotex and Gecko G203v drivers with good results. The HobbyCNC board will probably serve your needs just fine. Keep in mind that it's a unipolar driver. The 305 oz/in motors also seem well suited to your needs (in terms of speed and torque). Part of that depends on the drive mechanism, but it all seems like very reasonable choices.
Have a great time with the build, and document the process for others to learn from and enjoy.
Steve |