One of my friends was about your height and weight, and a couch potato. She took up cycling, just riding to work and getting around town. Within 6 months she thought nothing of doing a 70 mile ride. I wont say it turned her into a slim supermodel, but she was a much fitter and happier big gal.
Dont under-spend on a new bike. There are plenty of wannbe bikes, but a sensible fitness and general purpose utility bike is going to cost maybe $250+.
You can adapt your lifestyle to do more cycling. I do all my local transport and shopping on the bike, it pretty much pays for itself, and saves me time compared to other ways of travelling.