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.
What kind of hard plastic? And for what purpose? You can probably use a standard drill and drill bit
by using a drill
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.
A 1/4 inch bit will do it.
If you want the bolt to slide through the hole a 5/16 hole will work well.
The time it takes to drill a hole depends on the material being drilled, the size of the hole and the machine doing the drilling. But you knew that. It takes longer to chuck a bit in a drill than to actually punch an eighth inch hole through a piece of quarter inch plywood. But drilling a one inch hole through a twelve by twelve framing timber takes longer. Particularly if you're using a brace and bit. Are you drilling a four inch hole through a twelve inch thick slab of reinforced concrete? Roll up your sleeves and make sure you have a good water supply to lubricate and cool the diamond coring bit on your rig. Planning on punching a thirty inch hole through a thousand feet of mixed earth and rock to mount an emergency rescue effort of some trapped miners? Order out for some lunches. You and your crew will at this one for a while.
That's a pretty big drill bit, -it drills a hole just a tad larger than 3 3/8 inch wide.