Your Eurail pass covers most inter-city legs - just show it (or the Eurail Rail Planner app) on board, no ticket needed. Three legs need a separate paid reservation booked directly with the operator, regardless of the pass: Amsterdam→Paris (Eurostar), Paris→Basel (TGV Lyria), and Vienna→Rome (Nightjet sleeper). Book those as early as you can - they have limited seats/berths. Everything else (Basel→Interlaken, Interlaken→Zurich, Zurich→Vienna, Vienna→Budapest) is hop-on with no booking - just open the Eurail app for schedules.
Amsterdam & Rome & Budapest: tap your card or Apple Pay directly on trams/metro/buses - no ticket needed.
Paris: no direct tap yet - add a Navigo Easy pass to Apple Wallet instead.
Vienna: contactless isn't confirmed - buy through the WienMobil app to be safe.
Switzerland (Interlaken): no gates at all - it's honor-based. Use the SBB Mobile app (EasyRide) or your Eurail pass.
Eurail Rail Planner covers every inter-city leg. On top of that: Eurostar, TGV Lyria/SNCF Connect, and Nightjet for the 3 legs that need a separate reservation, plus WienMobil, BudapestGO, and Moovit/Citymapper for city navigation.