Unless your travel party includes an expert on the Normandy invasion, I highly recommend taking at least a one-day small-group tour of some of the key invasion sites. Most if not all the tour companies based in Bayeux use vans rather than large buses, so the driver/guide will move you around a lot more efficiently than you would do yourself. The guide will provide context to what you are seeing; with the exception of the museums in the area, there's not a lot of signage to help you understand the importance of the remnants of the war that remain in place. I was happy with my Overlord Tour, but there are other options; Rick mentions some in his guidebook, I believe.
If you have an extra day available (Bayeux itself can keep you busy for about a day; it has a good invasion museum as well as the cathedral, tapestry, and small historic center), a rental car would allow you to see some additional sites or return to places where you wanted more time. It's true that public transportation is quite limited, so you'd need to check bus routes very carefully before relying on that form of transportation. However, if you are flexible about what you see during you free time and are OK with choosing from among the limited places to which bus service is available, you can manage without a car. I didn't have a car myself. While staying in Bayeux I took the tour and tacked the loca sights, including the invasion museum.
Prior to settling down in Bayeux, I had spent several nights in the larger city of Caen, which offers better transit connections heading east. While staying in Caen I visited the large/costly Memorial de Caen museum (which I liked a lot but many others, preferring to focus just on the invasion, find overwhelming) and the museum covering civilian life (including the Resistance) located in Falaise. I also visited the coastal towns of Honfleur, Cabourg and Deauville. However, that itinerary--while possible by train and bus--is only suitable for those with a fair amount of time to spend in that part of Normandy. I'm a slow traveler and had nine nights to split between Caen and Bayeux.