Do you plan to rent a car? I did not, so I was somewhat constrained in getting around. I spent about a week in the Bayeux/Caen area. I took a one-day small-group (van) tour of some of the D-Day sites. Some companies offer multiple itineraries, so you could take more than one tour with little, if any, duplication of areas covered. I do think if you are not an expert on the subject, it's helpful to take at least one tour. Most of the D-Day tours depart from Bayeux. After taking a tour or two, there are some places reachable by public transportation you could see on your own.
Bayeux itself has a number of worthwhile sights: the historic center (not damaged during the war), the cathedral, the local invasion museum, the British war cemetery and the Bayeux tapestry. The tapestry will be going off display sometime this year for a change to the exhibition venue. I don't know the timing of that, but I'm sure it's readily available via Google.
In order to see other places in Normandy, I spent several nights in Caen, which has better transportation connections to places like Rouen, Deauville/Trouville, Cabourg and Falaise, where there's an interesting museum about the civilian experience during WWII, which also covers Resistance activities. Trying to see any of those places while staying in Bayeux would have required me to start each day with a trip to Caen, and it didn't make sense to me to do that over and over again.
Of the places I've mentioned, Rouen has the longest list of tourist targets, I think. It also has a very beautiful historic center (reconstructed after dreadful wartime damage).
Caen was plastered during the war and doesn't have a great deal of tourist interest. There's the very large (as in full-day commitment) Caen Memorial museum that covers the period leading up to WWII, the war itself and the Cold War. I liked it, but many people find that museum overwhelming, crowded and expensive. Also in Caen are the Chateau de Caen and two historic abbeys.
Mont-St.-Michel sits basically on the dividing line between Normandy and Brittany. There are also interesting towns in eastern Brittany, not far from Normandy, though some are quite touristy. To me these were worth seeing: St.-Malo, Dinard, Dinan, Fougeres, Vitre and Rennes.