Sports are a great way to keep your Lady in shape, so that she feels happy and self-confident. In the Gym you can let her work out for a certain time - 15 or 30 minutes.
Do not forget that even though sports are good for your Lady's weight, she will eventually get hungry and thirsty and her energy and hygiene will drop.
In order to use the Gym you need a membership card, which can be bought from the Market. You are allowed to have only 1 card at a time.
The difference between running in the park and working out at the Gym is that the Gym gives you bigger bonuses.