Trains from Bayeux

Each of the Paris train stations serves a different region of France (or other countries.)
Where do you want to go after Bayeux?

Our next stop would be Arles France

Then you will be changing trains in Paris if your next destination is Arles.

Unfortunately, there isn't one big train station in Paris. There are a series of train stations that handle traffic in a particular direction.

If you pack light, you can use RER/Metro trains to get from one station to the other. It's best to plan on an hour or more to change, especially if you have a connection or two. I think the last time I did it, it took me just about an hour to make my connection Gare de l'Est to Gare Saint Lazare. In your case, I believe it's Gare Saint Lazare to Gare Lyon.

You can also take a taxi from one station to the other, but it will still take awhile to get across town.

The train to Arles from Paris is about 4 hours. The train from Bayeux is another 2 hours. Plus time to make the change from one station to another. I'd at least consider a flight on a European carrier, it may or may not work better. As an example, you can make it from Caen to Marseile in about 3 hours (with a change in Lyon), 1 hour to Lyon, 1 hour layover, 1 hour to Marseilles.