Pilot Holes...
Hole Deeper than the screw... Say a Machine Head Screw, drill the depth from the screw point, to the underside of the head... when you secure the machine head there will be a fraction of hole left of the machine head thickness... Same for the bridge... just make sure the hole doesn't go all the way through. If the hole isn't deep enough, it will bottom out and you will strip the head.
As for hole size... drill should be the size (dia) of the shank of the screw, thats less the threads, to small and it will be too tight, rounding the head it, too big and there will not be enough bite for the threads... this can depend on the hardness of the wood, I recommend a try in a scrap piece of wood.
Screwdriver... Cross head screws differ... Pozi, Phillips and JTS.
Use a good fitting screwdriver i also use a layer of plastic sandwich bag between screwdriver tip and screw (bolt head to socket...) to protect the finish.