There are two different types of tools to do this. Ask for a 'spade bit' or 'hole saw' at the hardware store. Either one is used with an ordinary power drill. When cutting through plastic with power tools, do not run the tool at maximum speed as the friction may melt the plastic and make the cut edge less clean.
by using a drill
The recommended drill bit size for a 15/64 inch hole is a 15/64 inch drill bit.
The recommended size of the drill bit for a 7/16 inch hole is a 7/16 inch drill bit.
A 3/16 inch pilot hole drill bit should be used for installing a 1/4 inch screw into hardwood.
7/8 diameter drill or .875
For a clearance hole you need a 9/64 drill bit.
It depends on the material you are fitting it into. In most wood I pre-drill a 1/4 hole for this. In softer woods i would drill a smaller hole .
5/16 hole X 4 inch on a flat bottom hole: This will depend on the material, and tools available. Use a 1/8 bit, then 5/16. If you are not sure how deep the drill bit is in the hole, paint white out on the bit as your 4 inch mark so you can see it while the drill is spinning. If the material is metal, you will need coolant. To get the flat bottom: it wood, put a flat ended bolt into the hole and hammer, it will flatten the hole, it metal, forget about it, the hole is quite flat, and a pain in the butt to change. :-)
i would use a long shafted hole saw
One that's too big for you to hold.
You will have to drill a pilot hole before screwing into oak. Use a cordless drill and a 1/16" inch bit or 3/32.
To accommodate a 3/8 bolt, drill a hole that is 3/8 inch in diameter.