You'll get stamped when entering and when leaving the Schengen area. So 2 stamps at least!

Within these countries there are only random border checks and they won't stamp your passport.

Where are you travelling exactly ?

The plan is: Paris to London to Brussels to Frankfurt to Bern to Lyon to Paris. 
 

Two stamps is better than none!

Since London isn't in the Schengen area you'll get 4 stamps even!

Entry : Paris Airport

Exit : Paris Gare du Nord train station

Entry : London St. Pancras (for Brussels)

Exit : Paris Airport

I'll try to add a picture or two.

Side note : you should book the Eurostar under the Channel in advance. This route is really popular (it's by far the most convenient way between these cities) and there's a passholder quota -> unique in Europe.

Do not hesitate for further questions. :)

Thank you so much!! We are going early May so I’m trying to get everything in order. My mom is 83 and my sister has chronic pain, so their mobility isn’t great - my plan is to find a place to stay near the train stations, then do either a hop on/off bus or find a close-by main site to see, and eat somewhere fun. That should be sufficient for them to say they’ve “been there.”   Any recommendations are welcome.  I appreciate that above you indicated specific stations - some of these cities have multiple so I’m just guessing!  

Will definitely get the Eurostar under the Channel situated sooner than later - thank you for the tip!

Here are the pictures :

Note the little steam train icon. :)

As far as I know the stamps haven't changed much in the last decade.

Honestly Frankfurt isn't worth your time. Better to spend a night or two in Cologne or go straight away to Switzerland.

Don't be worried of train connections too, they're fairly common especially in Switzerland. Connections there are straightforward and there are plenty of trains.

The Eurail Global Pass is good value in the country. The pass is valid on boats on Lake Thun and Brienz, not far from Bern.

For the comfort and since it's a lifetime trip I wouldn't hesitate to get a 1st class pass.

Further questions welcome. Hope the trip goes well.

Good to know about Frankfurt - I’m really just looking to get the country check-mark, so I’ll look at Cologne.  We will have approximately 8 days (arrive on May 1, depart May 10)

Eurail Global Pass - first class pass! Noted! 

Can’t tell you how much I appreciate the information!  I’m sure I’ll have more questions/recommendation requests, so I’ll keep you posted! 

Okay with this limited time I'd rather head straight from Brussels to Switzerland via Germany. If you don't mind waking up early you could take a panorama carriage along the Rhine, beautiful!

More info here : https://www.seat61.com/places-of-interest/rails-down-the-rhine.htm

Itinerary would look as follows :

- ICE Brussels-Midi - Köln Hbf (Cologne) 08:25 - 10:15

- EC Köln Hbf - Basel SBB 10:53 - 15:36

- any train towards Bern (every 30 min)

Could you also fly out of Switzerland (Geneva or Zurich) to save travel time? Otherwise you'd have to spend the night before at Paris CDG Airport -> pretty annoying.

I'm not sure I'd add Lyon either for 8 days, it's your trip though of course. I'd spend more time in Paris, London and the Alps : 3 unique places!

They are worth each at least 3 nights in my opinion but you could go down to 2 if you must. Ideally I'd do:

- Paris 3

- London 3

- Brussels 1

- Bern and the Alps 2-3