To train for a season of golf, it actually makes sense to run. Competitive golfers don't just sit in a cart and drive from one shot to the next. Many low-handicap golfers will walk 18 holes and do it three or four times a week. Running 2 to 3 miles at a time can only help a golfer who walks 4 to 5 miles over a typical 18-hole round of golf.
Golfers get most of their power from their core muscles in their trunk. Lifting weights will help a core get somewhat stronger, but working with an exercise ball will help even more. Sit on an exercise ball and lift light weights with proper form or make small circles with your arms. This will help you develop a stronger core.
Back strength is very important to a golfer. Stretching out the back will only help a golfer with with range of motion and power. Lay down on the floor with your knees angled up. Put your arms at your side and lift your butt up off the floor 10 times. This will help your lower back.