Another method to consider is pocket holes. Essentially, the guide used
makes the process a two-part operation -- drilling the small hole for the
screw to pass through and a larger but shorter hole (partially into the wood
section) to take the screwhead below the wood surface. Yes, that's called a
countersink.

Screws go in at an angle, which gives them a bigger bite and more leverage
for a clamping action.

Anyway, check out Lee Valley Tools (based in Toronto Ontario) at their
website http://www.leevalley.com and enter pocket hole as the search term.
You should be looking at a half-dozen guides, drill bits as well as hole
plugs. Prices are in Canadian dollars, which convert to about $.75US.