You would just buy separate tickets between each city-pair. The Ams>Brussels, and the Brussels>Paris are on the Thalys line. Paris>Berne would be TGV.
If you hurry (with due diligence of course) you can buy some reduced-price tickets for August, they sell fast though (and have restrictions). Otherwise, full-fare tickets are exchangeable within parameters. All of these trains' prices include an assigned seat reservation, and all of these trains are specific dates/times that you buy.
No, you can't buy tickets like that that allow you to jump on and off. Each ticket is a separate journey, so you have to plan.
You're already really late buying tickets and the cheap ones will already likely have been sold, so buy them asap!
Book Paris to Bern at www.sbb.ch - from 25 euros on a direct train, more if there's a change at Basel.
Booking opens 92 days ahead and fares work like budget airlines - book early for the cheapest fares.
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