3/16" bit will do fine.Get screws that are 3/8" longer than the thickness of the wood.Make sure you buy or make a template for drilling the holes.HD sells some.

Get or make a template something like this:

template

Put tape over all the holes you're not using! (don't ask how I know this is important)

I use a 5mm Festool Centrex drill bit for cabinet knob screw holes, because it is a perfect size, and there is no tear-out on either side of the hole.

I used a brad-point drill bit and I eliminated tear-out on the back side (I call it the "exit wound") by clamping a piece of scrap wood behind the door.

The brad point was useful, because I could be absolutely sure that my bit wouldn't walk around when I started to drill; I could push that brad point into the wood a bit to anchor it.

Also--to mark the holes, I put blue painter's masking tape on the door and marked the spot w/ a pen, so I could mark them all and, check them all at once--very carefully--3 times before I actually drilled any of them. (I had unusual sizes of door pulls, so I couldn't just blithely rely on a template--but even if I'd had a template, I'd have done this sort of temporary markup.)

Take your knobs to the hardware store and see if you can buy longer bolts for them, because you're right, you'll need something longer.
(my drawers didn't need longer bolts, because I put the pulls on the top of the drawer front instead of in the middle; the drawer box was shorter than the decorative drawer front, so I only had to drill through the decorative front)

As for the thickness of the bit you should use, it depends on your bolt; I compared the shank of my bit with the bolt from my pulls, and drilled a test hole in a scrap to see how the bolt actually slid through. I switched bits because the first one I used was the tiniest bit bigger, and the bolt wiggled a little. The next size down had a nice, snug fit.