To burn fat, you simply need to eat less calories than you consume. So if you need 2000 calories, eat less than that and your weight (muscle/fat) will go down. Use resistance if you want to retain as much muscle as possible. That answers the burn-fat tag related question.
As for the cardio, short bursts of high heart rate will be fine. The gym is not the best place for cardio. It is very expensive if you're only there to run. Try to run or bike outside. A lot of people get bored running on a treadmill due to lack of stimulus. Outdoors offers better scenery if you're into that. If you live in the land of eternal winter, you are lucky. Save yourself some money and start shoveling your entrance, that's a pretty good work out.
Another option for cardio is HIIT (High Intensity Interval Training). HIIT will spike your heart rate during a short period of time and then give you a short time to recover before you start over. These sessions can be very short (Mine last about 12 minutes including 2min warm-up and 2min cool-down).
Discussing HIIT vs Steady State cardio is a different question.