My wife and I enjoyed Bath immensely. We were there 5 days. Two days were involved with the annual Jane Austen festival. My wife dressed in full regency costume for the festival. Transportation around the city is very good. You can easily get day trips to Stonehenge, Salisbury and other locations. (I really wasn't impressed with Stonehenge) but Winchester Cathedral was superb.
We did have a car for our day-tripping as we were driving to the Cotswold, the Lake District, Oxford, York and lots of other places. Parking is about 12 pounds a day. From Bath we did visit Hawkesbury, Lacock Abbey, Stourhead Gardens (very beautiful) and Castle Coombe (listed among the 100 most beautiful places in England. These places are a bit more difficult to get to without a car.
In Bath, we stayed at Three Abbey Green. Nothing better. We were 500 feet from the Roman Baths in a nearby very quiet courtyard. Also, the Roman Baths are located in a great open plaza next to a 4-5 block pedestrian shopping area. Great shops and lots of great food. The oldest house in Bath (circa 1400) is about 200 feet from Three Abbey Green. It's a restaurant.
For York, if your visit is 2 days and one night, you might be missing out if you want to visit Castle Howard (used in Brideshead Revisited). 2 nights and 3 days (including incoming and departing travel) will do it.
Send me a PM if you would like some pictures.