Fasten 1/4" steel plate to 2x4 wood?
The wood doesn't have holes


So I'd use something like a bradawl to mark the positions then drill a pilot hole if necessary (depends on wood, position of hole, diameter of screw etc)

See How can I drill to an exact depth with a hand drill?


  and I don't want the screw to go completely through the wood.


That's just a matter of buying the right sized screw.