What should the size of the holes be?
3/8" I think
What kind of bit makes the cleanest hole?
High speed steel bit, as long as it's sharp
Which is better, a high speed or a lower speed?
High speed, otherwise the bit will gall up and bind
Will a drill press give me any better results than a hand drill?
Use a drill press to get your holes lined up on either side of the barrel, I made a jig with some brackets to keep the pipe stationary while drilling
Miscellaneous?
Take your time and wear safety glasses
Good Luck!