You should get yourself a road bike. If you enjoy it, and if you have trouble sticking with a gym membership because their machines are stupifyingly boring, then it sounds like a good plan. Being able to change your scenery and visit your favorite parks, or other spots, play with inertia, feel the wind in your hair, and all that, makes it pretty easy to motivate yourself for a bike ride, and all the calories that go with it.

I won't tell you which bike to get, but your weight won't be a problem. I'd avoid sora shifters, if possible, but my girlfriend liked them on her last bike. Apparently once you figure it out, you can shift from the drops after all. And, you can always change the shifters out if you otherwise like a bike, but get tired of them - so I wouldn't let it stop you.

My road bike has a set of Fulcrum Racing 7 wheels, which I saw on bonktown for less than $200. I've put about 3,000 miles (almost 5,000 km) on them, mostly on decent roads but with some terrible pot holes, some curbs, and the like, and they've barely gone out of true. These are pretty good wheels. You can find other wheels that will hold your weight, too.