I suspect that "affordable private drivers"--by my definition of "affordable"--do not exist in France.
I have never been to the Loire Valley so I can be of no help there.
There are two basic ways to find bus schedules (and sometimes fares, but the fares are usually pretty low). One is to use Rome2Rio.com to look up each travel leg, then keep clicking through the website to find a link to the company operating each bus. It is important not to trust the fares, travel times or frequencies shown on the Rome2Rio website itself. They are often wildly off-base, and I consider them utterly useless. Actually, they're worse than useless, because they can lead travelers to make the wrong logistical decisions.
The second approach is simply to Google something like: Bus Caen to Honfleur. That's what I usually do. If Google doesn't deliver a link obviously belonging to a bus company, that's when I go to Rome2Rio. Rome2Rio is also helpful when you aren't sure whether you need to take a train or a bus, or where to change from one mode of transportation to another.
There is reasonably frequent train service between Bayeux and Caen. Most people would go to Caen to see the Peace Museum, which is not near the train station. There is city-bus service to the museum from a stop fairly near the castle (not terribly near the train station). I'm nearly certain that bus runs more often on weekdays than on weekends, and I'm not sure there's any service at all on Sundays. I have no idea how much a taxi would cost. Note that the Peace Museum is very large. One can spend a full day there. I wouldn't recommend it as a place to spend just an hour or two, because it will take considerable time to get there. Bayeux itself has a very good museum, and it's a more manageable size.
I know nothing about using public transportation to get to invasion sites, because I took an Overlord tour. I know there is some--probably infrequent--bus service to places that played a role in the invasion, but I don't know specifically where you can travel by bus. Aside from those sites, if you're staying in Bayeux, just about any trip you want to take in Normandy will start with a train to Caen. To get to Rouen or Deauville/Trouville, you would take a second train. To get to Honfleur you would take a bus from the Caen bus station, which I believe is right beside the train station.
Because of its role as a bus and train hub, I spent some nights in Caen as well as some nights in Bayeux. Because I also wanted to see Rouen, which I knew merited more than just a few hours, I opted to stay there as well. Even traveling from Caen, Rouen would have been a 3-hour round-trip, not including the time required to get to the Caen train station or to get from the Rouen train station to the historic center. We travelers--ever optimistic--tend to ignore those additional steps required for a day-trip.
You'll find that a day-trip from Bayeux to Honfleur requires quite a bit of time on a bus. It's probably still doable. Honfleur is rather small, so most visitors don't spend more than a few hours there.
The key thing to remember when planning public-transportation trips involving small towns is that service (especially bus service) tends to be infrequent, so it can be shockingly difficult to get to more than one such place per day. This is the way I travel, and I have become very accustomed to devoting an entire day to a trip to a town with only 3 or 4 hours' worth of sights. That doesn't really make sense for a traveler with limited vacation time.
I recommend being sure you're covering all the points of interest to you in Bayeux before setting out on an inefficent quest to see a distant place like Honfleur by bus.
