Most of the time paying for a gym membership is the best idea if you are trying to get fit - why? first of all you already paid a fairly big amount of money to be able to use the machines and beside of that there are a lot of people in the gym with same goals as you - so that can keep you going.
But as u said u don't have the time - so purchasing a home gym appliance can be the best choice.
Losing weight in one part of the body isn't as simple as it sounds. Our body is constructed so that it burns up calories from different parts of the body. This differs from person to person. The best thing you can do is get a gym appliance that gets your whole body working. A tread mill seems to be a good choice as it involves a lot of your muscles.
Ellipticals can be a good choice too but be sure to buy a good one - one that moves with your body in a natural way so that your moves are not forced - if they are this can give you more problems than help.
Beside that u can also buy a DVD weight loss program. U can get many good ones, from aerobics to workouts suitable for your living room.
Hope this helps!
Wish u a lot of success!