With the current sale a 10 day in 2 month pass would be the most cost effective.
This applies as long as your travel falls within the validity period for these sale passes so for a 10 in 2 month your travel would need to be concluded by 30 July.
I would recommend visiting Paris first and not at the end. You'll be exhausted after your flight and it's not a good idea to visit elsewhere right away.
Some advice about reservations :
They are mandatory between Paris and London + London and Lille or Brussels for Bruges (30€ extra). It's a popular route and there's a passholder quota so book in advance. Check availability on https://www.b-europe.com/EN/Booking/Pass#TravelWish
Reservations not possible (nor needed) in Belgium and the Netherlands as long as you avoid Eurostar to Amsterdam.
Between Amsterdam and Lucerne consider the good night train to Basel (early arrival)/Zurich. There are private compartments available if you book long enough in advance. With a pass you only pay accommodation fare.
It's also possible to do the journey by daytime trains but it would take most of your day.
Reservations aren't a thing in Switzerland : you simply log the train on the app and hop on.
Special case : 49 CHF mandatory reservation on the Glacier Express. Also possible to do the same route on reservation-free regional trains on the same route -> less crowded, pull-down windows for part of the journey :)
Unless you have a reason to visit the city Geneva isn't worth your limited time in my opinion. Add a night in London if it's your first time.
Geneva - Paris : 29€ mandatory reservation for the direct TGV. Can get down to 10€ if you cross the border on a regional train before hopping on the TGV.
With all that being said I'd get a 15 days within 2 months pass to include all your day trips. The price difference with a 10 day one isn't that big and then you don't have to worry about tickets for small journeys. 1st class also doesn't cost much more so consider it ! There's a 25% sale going on but passes must be activated on 1st June at the latest.
Further questions welcome.
Last thing : Interlaken is not a place to visit, it's a base for the surrounding region. You'll probably want to see Lauterbrunnen and/or Grindelwald. Railways to those places are not included from Interlaken Ost but you get a 25% discount, tickets available at all Swiss stations (valid all day). Otherwise full-fare on SBB app or sbb.ch.
On the other hand boats on Lake Thun and Brienz are fully included ! Feel free to take them, they're a great experience.
Lauterbrunnen and Grindelwald are so small that they'd take you half a day at best so by mid-afternoon you could be on a boat from Interlaken to Brienz before heading to Lucerne ! :)
Very helpful responses everyone!
I’m confused though about the suggested 10 or 15 day passes since we will be travelling by train pretty much every day of our 21-day trip. How does 10 or 15 travel days cover all the other days we will be travelling trains/subways to do day trips to museums/visit or get to and from our accommodations, e.g. from Montreux to visit Gruyere or take a day trip from Paris to Dieppe?
Many thanks!
I’m confused though about the suggested 10 or 15 day passes since we will be travelling by train pretty much every day of our 21-day trip. How does 10 or 15 travel days cover all the other days we will be travelling trains/subways to do day trips to museums/visit or get to and from our accommodations, e.g. from Montreux to visit Gruyere or take a day trip from Paris to Dieppe?
You've listed only 9 journeys and a few day trips. We can't assume there are any other trips if you don't tell about them. If you'll be on the train almost every day, then maybe a 21-day pass will be better.
However, only trains are included, not metro, tram or bus (with only few exceptions).
