transportation from Bruges, Belgium to Normandy, France

Normandy is pretty large. Where in Normandy?

Rail is probably going to be your best bet. You can check routes, times, and costs at tgv-europe.com, a site of the French national railway. Just keep setting your country to Canada or Great Britain (if you set it to the US, you'll be kicked over to the RailEurope site), input Bruges and your destination, and routes and costs will come up.

The French railway network is very biased towards travel between Paris and the regions, rather than region to region. You'll probably be fastest to travel from Bruges to Brussels, then by Thalys to Paris, and then from Paris to wherever you are in Normandy. You will have to cross Paris by Metro: trains from Brussels to Paris arrive at Gare du Nord and most of Normandy is served by Gare Saint-Lazare.

If you want to have a car to explore Normandy, take a train from Bruges to Lille and pick up a car there. That way you will avoid the hefty fee charged for picking up a car in one country and dropping it off in another.