Although there are exceptions, most of the D-Day van tours depart from Bayeux, which is an attractive small town. If you're taking one of those tours, I think Bayeux is the best base for at least part of your week. I think you make Caen work (nearby, and with train service), but I didn't want to have to worry about commuting into Bayeux for the early starting time of my tour. While staying in Bayeux you can also see the cathedral and the Bayeux tapestry, plus the WW II museum (see below). Bayeux wasn't significantly damaged during the war, so it retains some lovely historic architecture.
I enjoyed my Overlord Tour very much, and others have also been pleased with that company. But there are other options, and I don't think I've read a negative review of any of the local companies that run small-group tours. I know there have been multiple positive comments about Dale Booth as well as Overlord, for example. Overlord has more than one tour itinerary, so it's worth checking that company out if you think you might want more than one full-day tour.
The Peace Museum in Caen (bus from city center or taxi) is huge and is not limited to the Normandy invasion. I spent essentially a full day there; some others find it overwhelming and prefer more-targeted museums.
The WW II museum in Bayeux is good and more manageable in size.
The Airborne Museum in St-Mere-Eglise is something I was able to just about see in its entirety during the Overlord lunch break in that town, but I am not interested in war materiel such as planes, so a hardware junkie would probably wish for more time. This museums is fairly small and has a less-modern feel to it. I found it the least interesting of the four museums I visited, but it depends on what your focus is.
There's a relatively new museum about civilian life during the war (also covering the Resistance) in the town of Falaise. I visited that museum while I was based in Caen, traveling by bus. The town was basically leveled during the invasion, so there's not a lot to see besides the museum itself.
Caen is a transportation hub and is a workable base for coastal towns to the east (Cabourg, Deauville/Trouville, Honfleur). I stayed there--despite its proximity to Bayeux--because it is also connected to Falaise and I knew I would be spending a lot of time in the Caen Peace Museum. Otherwise, to split time between two towns as close together as Bayeux and Caen seems silly.
Note that some of the coastal towns east of Caen have only bus service, but the train stops at Deauville/Trouville. The bus service is fairly frequent, but it varies by day of the week, so you'd want to check the schedule carefully. There's also an express bus to Honfleur.
I also stayed in Rouen to see the sights and museums in that city. The (rebuilt, post-war) historic district is large and quite beautiful, there are at least three museums, and then there are the Jean d'Arc-related sites. It would take more than a full day to see everything, but a half-day stop on the way to or from Paris (if you can check your luggage somewhere) would be worthwhile if you don't have time for more. You'll have to be brutal in selecting what to focus on, though.