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.

You never book a reservation free train.

In the Railplanner you can add a filter that only shows the reservation free trains. This should work for countries that share their full time table with Eurail. However, the information in the Railplanner is not always correct so the advice is to always double check information with the national railways.

Indeed, no need to “book” anything. Just put your journey in the pass and generate the code. That’s your ticket on you’ll be able to jump on any (reservation free) train. The options @EdM gave are very good, but they take time. Much more, but if you like trains and a bit of adventure along less known places, it’s very enjoyable.

There are two direct TGV trains from Brussels to Strassbourg as well. They are usually well filled with Eurocrats travelling between the two EU parliamentary cities. But you’ll need reservations of course.

Domestic trains in Belgium are not reserveable. Just show up on get on them with a valid pass. On weekdays there are three trains an hour from Bruges to Kortrijk.

When are you planning to travel?

Please note that there are engineering works between Moeskron/Mouscron and Lille Flandres this semester. (This is the route your reservation free train takes). Some trains get replaced by busses.

More info is only available only in Dutch in French: https://www.belgiantrain.be/en/travel-info/current/ongoing-disturbances-and-works. If you state your planned itinerary we can help figure you out the reservation free options - without busses.