See above?mmmmmmmmmmmmmmmmmm
The main problem? The sea in the middle. There’s no direct train from Southampton to France so if you want to take the train you’ll have to go back on yourself to London, then catch the Eurostar train to Paris.
Train from Southampton to London Waterloo:
Tube from London Waterloo to London St Pancras: no need to prebook.
Eurostar train from St Pancras to Paris: www.eurostar.com
Train from Paris to Bayeux: not sure
Alternatively you could fly from Southampton Airport to Paris.
Have you studied the Main in Seat 61? He is the absolute expert and will likely answer all your questions. Of course, I'd likely stop in Paris on the way, because...well, why not, it is Paris! This sounds like a fun train trip.
Just to add to the previous posts
buy the Eurostar tickets as early in advance as possible for the best prices. Arrange your trip into London, and allow sufficient time to arrive at St Pancras AT LEAST an hour before the Eurostar departure time. You have to clear both UK exit and France entry passport control before boarding.
in Paris you need to get from Gare du Nord (Where Eurostar arrives) to Gare St Lazare ( where the train to Bayeux departs). It's easiest, with luggage, just to take a taxi. But there are hustlers at the station. Only take a taxi from the official taxi stand outside the station. And insist on the use of the meter.
buy your ticket to Bayeux in advance. Either use the official website: https://en.oui.sncf/en/train-ticket
OR, if you have problems with your credit card being accepted (I'm seeing numerous posts from Americans with this problem), use a ticket site like loco2 or train line.eu.
If you want to take a ferry instead of the train, and are willing to go to Portsmouth instead of Southampton, you'll find departures for a few different channel ports. The nearest is Caen, but Cherbourg is also not too far. There are trains to Bayeux from both Caen and Cherbourg, according to Rome2Rio.https://www.rome2rio.com/map/Cherbourg-Octeville/Bayeux-Normandy-France
For ferry information, see this site. If you type "Portsmouth" into the "Outward Route" field, you'll see a drop-down menu of landing ports. https://www.aferry.com/portsmouth-ferry.htm?msclkid=0bb2678c4921104ca9243acc3d8722db&utm_source=bing&utm_medium=cpc&utm_campaign=COM%20-%20AFerry%20-%20Destinations%20-%20Desktop&utm_term=ferry%20portsmouth&utm_content=Destinations%20-%20Portsmouth%20-%20Exact
Be aware that not every route is served every day. Some boats rotate on different routes, or go out of service for maintenance.
From Southampton, I'd really aim for one of the ferries from Portsmouth (ideally to Cherbourg as it's quicker, but Caen also works - Le Havre is too far away), or Poole near Bournemouth for the Poole-Cherbourg crossing. You then take the train from Caen or Cherbourg to Bayeux, there is no need to book these trains in advance but you do need to book ferries and study schedules to see what suits you.