here is what I do.... first I punch-Drill the grid, in my case using a template. I do this because I make all my block in this fashion and for mass production a template works much faster. You would probably use a punch and hammer, to transfer the grid centers onto the copper.
As in this picture, illustrating the locations printed out onto paper.....
after that, I use a drill bit about 40% in diameter, of the final hole required. This drill bit is used to pre-drill all the holes to depth and act as a guide to the final bit. you will find it much easier and your accuracy will greatly improve, even though it might take a bit longer, since you need to run through all the holes twice.
other than that, do not be hasty.... you are busy with artwork, and all good things take time..... heheeh ya right.. look who's talking....