Most of the D-Day tours depart from Bayeux. If you intend to arrange a private tour, I suppose you'd have more flexibility. Caen was obliterated during the war and rebuilt in modern style with few vestiges of pre-war architectures and a distinctly finite list of places of interest to most tourists--a couple of abbeys, the castle, and the large/time-consuming/expensive Caen Memorial Museum, about which opinions are very mixed. (I liked it; many others find it overwhelming and too broad in its coverage, starting as it does with the pre-war period and extending into the Cold War era.) The Memorial Museum is well out of the center of the city, but there is public-bus service. Caen, despite its limited number of tourist attractions, is from the logistical perspective an excellent base for side-trips via public transportation to places like Honfleur, Falaise (interesting newish museum about civilian life, including the Resistance, during the war), and beach resorts such as Deauville/Trouville and Cabourg.
Trying to visit the places mentioned above (other than Caen itself) from Bayeux via public transportation would be a bit frustrating, I believe, because you'd have to add on the rail leg between Bayeux and Caen. It doesn't take long, but that probably will leave you cooling your heels for some time in Caen as you travel in both directions.
Bayeux does have some points of interest of its own, so allow yourself enough time there to do some things in addition to taking your D-Day tour. There's the tapestry, the cathedral, the Memorial Museum of the Battle of Normandy (modern, very good, and much more manageable in size than the Caen museum), a British war cemetery and the attractive (not terribly large) historic center. I took a walking tour offered by the tourist office that was good (as such things usually are).
Public transportation from Bayeux to individual invasion-related sites is limited but not totally non-existent. I believe there are others on the forum who did some invasion-related sightseeing by taking advantage of local buses, but I did not and cannot supply any specifics. I imagine Rick covers this topic in his guidebook, but I would confirm any information you will need to rely on, because COVID may have changed things.
It isn't impossible to get to Mont-St-Michel without a car, but a car will certainly make it a lot easier. The Churchill Hotel in Bayeux used to (pre-pandemic) run a shuttle that made a day-trip possible. One did not have to be staying at that hotel to take the shuttle. That shuttle does put you at MSM during the typical day-tripping hours when it is, by all (pre-pandemic) accounts grotesquely and most unattractively crowded. Thus the recommendation for arriving around mid-afternoon and spending the night so you can see the abbey when it is relatively peaceful.
There is rail service from Bayeux to the station nearest MSM, which is Pontorson, then there's a sort of shuttle bus that goes to MSM. But neither of those services is frequent. If something happened that caused cancelation of the train you were relying on, you'd be up the proverbial creek.
I believe there may be some commercial tours from Bayeux to MSM, but again (like the Hotel Churchill shuttle service) you are dumped at MSM at the same time all the other day-trippers arrive, which (it sounds like) seriously degrades the experience.