I would like to know what options I have to get around some cities and towns in France or Italy where there is no train service, without renting a car?
Generally, you have few to no options but it would be impossible to advise without knowing specifically to what cities and towns you are referring.
Buses are often a perfectly viable option, especially for traveling from a larger town to a smaller one within the same region. It definitely does depend on your desired route. Even when dealing with places served by trains, crossing regional boundaries can be a challenge. In France, for example, you may need to travel into Paris and back out again, which makes the trip a lot costlier as well as more time-consuming. It was my experience that buses sometimes went more directly where I wanted to go than trains did.