Hello Vie
Welcome to MySwissAlps.
Please get out a good map of Switzerland. Go right to the centre of the country and locate the city know in English and French as Lucerne, but called Luzern in German (the language spoken by most of its inhabitants).
I have been visiting Switzerland regularly for fifty-five years On my very first trip I based myself in Lucerne. On my second visit I based myself in Interlaken.
Lucerne (https://www.myswissalps.com/town/lucerne/) is a very historic city while Interlaken is a creation of late nineteenth-century tourism.
Lucerne is an ideal base for exploring the whole of Switzerland.
Interlaken is a good centre for exploring the Berner Oberland region (https://www.myswissalps.com/travel-ticket/berner-oberland-pass/), but not for longer distance excursions to places such as the Rhine Falls.
The boat service on Lake Lucerne (https://www.myswissalps.com/activity/boat-lake-lucerne/) is very good. Those on Lakes Thun and Brienz are not as good.
INVEST in a six-day Swiss Travel Pass (https://www.myswissalps.com/travel-ticket/swiss-travel-pass/). This will give you the freedom of the world’s best public transport system. It includes the buses, trams/streetcars and lake boats, as well as trains.
The STP usually does not give free travel on lines which simply go up mountains. However in the Lucerne area it does give free travel on the lines up Rigi, Stanserhorn and Klewenalp.
In the Interlaken area the STP will take you to Lauterbrunnen and Grindelwald, and also up from Lauterbrunnen to the mountain villages of Mürren and Wengen. (On the mountain lines above those villages; you will get a discount.) (Lauterbrunnen is the village for waterfalls!)
Lauterbrunnen and Grindelwald from Lucerne.
The highly scenic Luzern-Interlaken Express runs every hour and takes one hour and fifty minutes. At Interlaken Ost you change to the local trains of the Berner Oberland Bahn. These trains run every 30 minutes. The front half of the train goes to Grindelwald, the rear half to Lauterbrunnen.
Rhine Falls from Lucerne
The Rhine Falls (https://www.myswissalps.com/activity/rhine-falls/) are close to Schaffhausen, in the north of Switzerland. To get to Schaffhausen you go via Zurich. After viewing the falls I would suggest you go on to the small medieval town of Stein am Rhein. Return to Lucerne via Winterthur and Zurich.
Other Excursions from Lucerne.
Take the Voralpen Express (https://www.myswissalps.com/activity/voralpen-express/) to St Gallen via Rapperswil. From St Gallen take a local tram-train to the splendid village of Appenzell.
Take the Treno Gottardo along the OLD Gothard railway, which goes ‘over the top’ via Airolo and finishes in Locarno, in the Italian-speaking part of Switzerland.
Arriving in Switzerland from France.
Coming from Paris get the TGV high-speed train from Paris to Zurich, but go only far as Basel. There are frequent direct Swiss trains from Basel to both Lucerne and Interlaken. Your STP will be valid from Basel to Lucerne (or to Interlaken).
If you are coming from southern France, travel to Geneva, (probably via Lyon). There are hourly IR15 trains from Geneva to Lucerne. This journey takes three hours. STP valid (of course) for all your travel in Switzerland east of Geneva. (For Interlaken you would need to change at Bern.)
Travelling to Switzerland from France: https://www.myswissalps.com/travel-guide/france-switzerland/
Best wishes for your detailed planning.
Roger