Screwing project to wood background

.....Yes, I have had to drill holes into resin.....

  I have drilled everything from from a pin vise size to the smaller dia. drill bit size.  What I do is use a slower r.p.m with the electric drill with no downward pressure....let the weight of the drill and bit do the cutting.  If the hole that needs drilled is bigger than 1/8th" I would suggest starting with a smaller pilot hole and go up till you get the hole size that you need.