Drill a hole trough a large wooden sheet first and use it as a guide. Have someone hold the board in place.
Don't try to force the drill to hard, and let the hammering do the work

Wood screws are fine for this. Match the size to the hole in the metal plate, or a bit smaller. Since you're probably screwing in thin wood trimming (meranti wood?) You can never go wrong with drilling a pilot hole. Use a drill that's a bit bigger than the shank of the screw in this situation.

I know you have drill bits, but if you're drilling in sheetmetal (hinges), I might advise on using a step drill bit.
These work alot better on drilling holes in thin metal and enlarging holes. Twist drill bits tend to "grab" sheet metal, and when enlarging holes break off at the small cutting area (dulling).