Paris to Normandy to Loire

Paris to Normandy to Loire can be done by train, but whether it can be done "easily" you'll have to judge.
What's probably not feasible is starting in Paris, doing a D-Day beaches tour, and then getting to the Loire Valley, all in one day, and by train.

First, there's no train to the D-Day beaches. None. There are bus tours from Paris, you could find one that would take you by bus from your Paris hotel, do a fairly rushed tour of some of the D-Day sights, and drop you back at your Paris hotel or close to it.

Most travelers going to the D-Day beaches by train sign up for a day tour (look at the Overlord tours website as one example) but those tours generally leave from Bayeux and want you to be in Bayeux fairly early in the morning to hook up with the tour. So most travelers sleep in or near Bayeux the night before the tour. You'll be returned to Bayeux in the evening, but then you have to get to the Loire Valley. And there you are in Normandy, and you need to get to your Loire Valley train stop. And most trains from Normandy go back through Paris, then to a Loire Valley train station.

I think you can see the challenges. It's doable but very likely not in one day.

I don't bike myself, but I wonder if you have done enough work on what daily distances are involved in your Loire plan? Do The kids have open (with cars) road experience? Can you rent bikes for them that can keep up with the adult bikes? Is this trip summer 2019?

I admire the interest in Normandy, but you need to evaluate whether the beaches are sufficiently interesting and rewarding for your children's level of knowledge of WW II. (I was a big reader at 11, but I understood little about wars.) You need to avoid the actual anniversary dates, too. We spent three nights in the Loire and three nights outside Bayeux, but we had a rental car. Are you aware of Mont St Michel?

We delayed the Loire and Normandy to our sixth visit to France. It was a lot of driving.

We have been to both the Loire Valley and Normandy. In the Loire, we stayed in Blois. It is perfect. Not expensive and managed to see the Chateaux that we wanted to see.

Normandy is great, consider staying in Bayeux. The home of the 900 year old Bayeux Tapestry that tells the story of the Norman Conquest of England in 1066. It is amazing, don't miss it.

You could do both Normandy and Loire in 4 days, but it would be too much in my opinion. Plan on coming back.

Generally on weekdays (but you must check your actual travel date) there are trains between, for instance Caen and Amboise that require only one connection at Tours or St. Pierre-des-Corps and avoid going into Paris (which automatically adds a connection between stations by metro or taxi). I see these around 9:00, 10:45, 12:50 on my test date, and they take 3.5 hours. These trains originate at Caen, so starting from Bayeux requires an extra connection.

Looking Up Train Schedules and Routes Online gives you the Deutsche Bahn train schedule link and tips for using it (but doesn't sell ticket for France). Also see https://en.oui.sncf/en/ for the French perspective, with pricing.

We visited both of these destinations, but we had a rental car. It was still a long haul. With 4 people your train fares will add up, so you might want to do the math and see if renting a car is worth it. I'd recommend getting the car at CDG so you can get right onto the major highways and not have to stress in Paris surface-street traffic. Driving from Normandy to the Loire region, watch out going through LeMans as two Autoroutes run together for a few miles. When they separate out again, you want to be heading toward Bordeaux (south), not Rouen (north).

Riding bikes around the Loire Valley is a popular activity, especially in the region around Amboise where the chateaux are a little less spread out. Even there, you'll need to allow plenty of time on the road from one chateau to another. Are you OK with riding bikes on traffic-filled roads in bad weather? Because the chances of rain are significant.

A problem with having a rental car to visit the D-Day Beaches is: you don't really know what you're seeing or where to go and where not to use up valuable time--unless you do a significant amount of research by yourself. That's why many travelers sign up for the D-Day day trip tours.

Learning "what you're seeing or where to go" is simple enough to do when you are at home before your trip—rather than wasting valuable vacation days. Even if you choose to do a tour, the more you know ahead of time, the more you will get out of it. Read a couple of books, watch a couple of movies, do a wiki search, and you'll be much better prepared.

That being said, if you don't want to rent a car you almost have to book a tour. This is a fairly rural area, and none of the sights are convenient to public transportation at all.