Running and jogging is an easy and convenient way to get in shape and meet people. Running is simply jogging FASTER. Running virtually no equipment and can be done anywhere, and can be done alone or in groups. If you have concerns about your heart or breathing discuss it with you doctor first.
Running shoes (sometimes called "cross-training shoes") should be both comfortable and lightweight and provide good arch and ankle support. Every foot and every shoe is different and you should try on the shoes you buy and even walk around for a few minutes to ensure a comfortable fit. Some optional items to consider are: a wristwatch/stopwatch, a water resistant wind suit (Gortex is best) for all-weather running, and a water bottle. Almost any sporting goods store sells the equipment, and there are several sports shoe even running specialty stores in the city.