Please note that you are planning to visit Switzerland, not "Swiss." Swiss is an adjective in English; Switzerland is the name of the country in English.
It is a little confusing, because the name of the country in French is "la Suisse" and, in German, "die Schweiz."
From long habit, I usually choose to use "Luzern" for the spelling of the city, because that is what you will see on road signs and in train stations in the region; it is the Geman spelling, and Kanton Luzern is a German speaking region. Strictly speaking, I suppose, in English and French I should use use Lucerne, as you did originally.
<<"Zurich, not specific but Rhine falls, Zurich Zoo, Lake Zurich etc.">>
If you have to chose between visiting Lake Zürich and Lake Lucerne I'd choose Lake Lucerne.
(In German, which is the language also spoken in Kanton Zürich, you would say der Vierwaldstättersee, literally, the Lake of the Four Forest Cantons or less common, der Luzernersee. "Der See" is "the lake." in German)
Lake Lucerne is surrounded by mountains and has more varied scenery than Lake Zürich.
In March, the lake boats will still be on the "Winter 2016" schedule with limited service:
www.lakelucerne.ch/fil eadmin/lakelucerne/con tent/Dokumente_Downloa ds_PDF/01_Fahrplan-Preise/01_Fahrplan/Sai sonaler_Fahrplan/2017/ 2017_Fahrplan_Winter.p df
However, the main route that runs the full length of the lake from Luzern to Flüelen does run three times per day. You can get off at a some places such as Vitznau, to go up on the Rigi on a cograil train, or Brunnen or Flüelen to catch a train back to Luzern.
The Zoo in Zürich is very nice, but not very large. As such, it does not offer much "extra" versus what you might see at other zoos around the world, or in Basel, where the Zoo is much larger. One of the nicer exhibits at Zürich is the Asian elephant exhibit. Perhaps you are already familiar with elephants? ;-) Of course, if you have children in your party, a zoo is usually for the childen a welcome change from looking at boring scenery and old buildings. ;-)
I am the wrong person to comment on the Rhine Falls:
Many people enjoy them, and in late March, the snow melt will be building volume for the water flow. I have seen a few water falls here and there, and visited the Rhine Falls once. We have not gone back.
We have gone back to Luzern many many times. Adults and children with me have never failed to be impressed by the Bourbaki panorama:
commons.wikimedia.org/ wiki/Category:Bourbaki -Panorama
For many, the Transport Museum (das Verkehrshaus) is worth a look:
I always enjoy a walk around the old town in Zürich, on both sides of the river, and many like to walk the well known Bahnhofstrasse for shopping. Opinions about old cities are very personal, but, I like the one in Luzern better. Since i always stay in the Niederdorf (part of the old town on the east side of the river, ) in Zürich, I usually walk around and enjoy it.
I can't help with material on rail passes. I don't know the details. Arno and Annika are the experts. I'm sure that they will jump in.
Luzern and Zürich are connected by frequent train service taking about 45 minutes. It is a nice ride, with views of the mountains near Luzern and the lake near Zürich. So, you can stay at either one and see the other, too. And, Zürich airport has good train service from Luzern. Among the trains, there is one every hour at xx10 that does not require a change of trains in Zürich Hauptbahnhof (main station). Easier to use if you are hauling a bunch of luggage. I do not know if you are flying or taking a train to Venice......
Rail service from Switzerland to Italy has several routes - two of the main lines fromthe Zürich/Luzern region are - Basel- to Luzern to Arth-Goldau to Chiasso to Milano Centrale, or Zürich to Arth-Goldau and thence to Chiasso. The lines cross at Arth-Goldau, and transfer between the lines is easy. The trains are stopped on opposite sides of the same platform.