The first thing that I'd recommend doing is making sure that you're following the procedure closely. Sometimes in science, simple mistakes can cause an experiment to not work correctly; these are easy to catch with a quick review, though.
For this experiment specifically, I have a few suggestions. First, I'm not sure exactly what you mean when you say that mixing red and blue doesn't cause any change- there should be a different color where the two beams cross, though the colors outside of that area will still be different. Pointing flashlights of different colors at a spot should produce an effect like the one in this picture:
What exactly are you seeing where you cross the flashlights? And when you shine one flashlight on white paper by itself, does it show the correct color?
Good luck, and hope this helped a little!