I will mention trains with affordable reservations as well as without, because your research may have previously pointed you in the direction of very overpriced international TGVs.
Brugge to Paris:
First, take the train to Kortrijk, then another over the French border to Lille Flandres (when you are in Kortrijk, Lille will appear on station departure boards under its Dutch name, Rijsel).
Then either:
- Take a direct fast TGV to Paris, making no stops and taking only an hour (reservation €10, no booking fee if you buy through SNCF’s secure international hotline 00 33 1 84 94 3635). OR
- Take a TER (regional train) to Amiens, and then another one on to Paris. Of course this will take substantially longer.
Paris to Strasbourg
- Pay €10 to reserve a seat on one of the frequent TGVs, taking around 1h¾, OR
- On Saturdays: take the direct TER leaving Paris at 08:35
- On Fridays and Sundays: direct TER leaves 16:35
- All other days: take the 16:35 TER and change onto another one when it terminates at Sarrebourg.
Strasbourg to Zermatt
Take one of the frequent fast TER services to Basel, then change onto the Swiss rail network on which (apart from on certain tourist trains) reservations are not required.
From Basel there are direct trains to Visp (from which the narrow-gauge railway to Zermatt leaves) every two hours, but there are also other itineraries with changes. If you go Basel - Spiez - Brig - Zermatt, you can take the old line over the Loetschberg Pass, which offers huge and spectacular views on the descent into the upper Rhone valley.