I want to go from Brussels to Bruges on Friday Oct 19. Then from Bruges to Bayeux on Sunday Oct.21. Bayeux to Paris on Oct 24. Should I make train reservations on line at home or wait until we are in Europe?
Wait until you're in Europe - it will be cheaper and you won't need reservations for most of the trains, especially since it's not peak travel season. There are several running daily to Bruges as well as Bayeux to Paris. I'm not sure about Bruges to Bayeux, but if it concerns you, make a reservation for Bruges to Bayeux when you arrive at the Brugges train station, etc.
I'd say wait. Your Brussels to Bruges connection won't require a reservation and the others depend on which connections you take, but in October you will be fine. Just make them from Brussels when you get there.
For Bruxelles-Bruges, there is no advantage to booking in advance.
For Bruges-Bayeux, see my post on your other thread.
For Bayeux-Paris, discount fares are available if booked far enough in advance at www.sncf.com.