In addition to the trolley mentioned above, there is city bus service and plentiful cabs. Also, if you select a hotel in New Town, the Old Town Trolley serves that area. My sense is that trolleys are not a great choice for day to day getting around, although a tour is highly recommended.
Bike riding is relatively easy, especially if you stay off major streets and absent that, cabs are going to be the most convenient/easiest physically/better price than a trolley. Island Comfort Mobility is also a resource for assistance with mobility challenges and might be able to hook you up with a mode of transportation that works for your situation.
My suggestin for your stay is to find a spot in Old Town near to some of the places you most want to see and plan to visit. Somewhere near the Historic Harbor Walk, perhaps? Hotels like the Hyall, Pier House, Ocean Key and Westin are good places to start looking.