Savage's approach to research, via Mosteller:

As soon as a problem is stated, start right away to solve it. Use simple examples. Keep starting from first principles, explaining again and again what you are trying to do. Believe that this problem can be solved and that you will enjoy working it out. Don't be hampered by the original problem statement. Try other problems in its neighborhood; maybe there is a better problem than yours. Work an hour or so on it frequently. Talk about it; explain it to other people.