How many calories can I burn exercising for ½ hr per day, 3-4 times a week?    How do I decide what gym equipment will help me most?
This depends on what mode of exercise you choose to do, the intensity at which you perform the exercise and how much you weigh.  For example, in 30 min. you can burn approximately 400 calories doing step aerobics, 250-500 calories bicycling, 300 calories on an elliptical runner, 180 calories walking and 300-400 calories running, all assuming you weigh 150 pounds.  If you weigh less than that, you will burn fewer calories; if you weigh more than that, you will burn greater calories.

I recommend interval training; for example, 10-15 min. of 3 different exercises to keep everything fresh.