Your 3/8 drill should be strong enough. That is if you mean the drill and not the bit. I use an 18v 1/2 inch drill. For a hole in the side wall of the bucket, I use a several bit sizes starting with 1/4 inch.
Take your time, use sharp bits, keep the drill speed low (around 200 RPM) and don't apply too much pressure. Your hands should not get tired while holding the drill in position. A slow drill speed will almost let you see the flutes in the drill bit as it turns. Plan to use about 4 to 6 bits to work to hole up to 3/4 or 1 inch size. It took me about 5 minutes per hold to do it in the side of my bucket.
As the hole gets bigger than 1/2 inch, be careful; the larger drill bits will snag the edges of the hole and that hand drill will drill will jump violently. Stop, back the bit out of the hole, hold on tight and change to full speed. Then move the bit into the hole slowly without much pressure. The edges of the bit will cut the edges of the hole.
The chips that come off of the bit can be shaped like the number '9' or a small ribbon of material. You don't want to see little flat flakes when running at low RPM.
