That's an excellent service business to have. Many senior citizens need transportation to and from medical appoinments. A few things you'll need to look into may include: business liability insurance of at least one to three million dollars (which really isn't as expensive as it may sound), vehicles capable of transporting disabled persons, and a comprehensive advertising plan.
You may consider working for a company that specializes in that business first to see if it's a good fit for you, then be careful about a "Non-Compete" waiver they may want you to sign, stating you will not take their customers away from them. They'll probably have that whether or not they know you are interested in starting your own service.