Basel to Swiss Alps to Frankfort

Susan,
Basel to Luzern is just over an hour from Basel by train, and there are two departures every hour. You just go to the station, buy a ticket and get on the next train.
You can look up train times and get prices on the SBB (Swiss Federal Railways) website: https://www.sbb.ch/en

If you want to go to the Berner Oberland, for example Lauterbrunnen is 2½ hours from Basel, again two departures per hour.

In fact all rail lines in Switzerland, and all stations, have at least one train per hour in each direction, at the same minutes past every hour.

The Glacier Express express runs east-west: Zermatt - Brig - Andermatt - Chur. If you want to do this you need an overnight, It is difficult to do it from Luzern as a loop all in one day.
This is a standard rail line, one train an hour, so you can get off and back on later. The Glacier Express is an extra tourist train with fewer stops.

Which starting point is recommended if at the end of our train trip, we will head to Frankfort to rent a car for traveling through the German Palatinate region?

I don't think you mean Frankfort, that is in the USA. I guess you mean "Frankfurt am Main".
You start wherever you happen to be. There are 1838 stations in Switzerland, you can start from any one of them and get to Frankfurt (Main) by train. Again look up (wherever) to "Frankfurt (Main) Hbf" on the SBB website.
"Hbf" = Hauptbahnhof = Main station, the one in the city centre used by long distance trains.

If you want to visit Lucerne and the Alps you might also consider the Gotthard Panorama Express boat/train that goes the length of the Lake of Lucerne on a steamer and then a train over the twisty old Gotthard Pass train route, ending in Bellinzona or Lugano....It's about a two hour ride back to Lucerne through the high speed new Gotthard tunnel so it can be done in a long day - or have a nice night in Ticino - the Italian part of Switzerland - they left in the late 1800's and came back...

Great Places!

The Glacier Express journey takes 8 hours. You can see the schedule here:

https://www.glacierexpress.ch/en/travel-planning/timetables/

Personally I think that is too much time to spend on a train; I would rather be outside with my feet on the ground innthe Alps.

The Gotthard Panorama Express to Lugano suggested above is a good idea. You could spend a night there and then return by a different route, such as the Bernina Express.

Here is a rail map of Switzerland to help you figure it out:

https://www.sbb.ch/content/dam/sbb/de/pdf/freizeit-ferien/ferien-kurztrips-schweiz/internationale-gaeste/sts-geltungsbereich.pdf

Thanks to all of you for your responses. I will research further and still welcome any additional suggestions.