Help with Zurich/Zermatt Itinerary during December

If your flight is from Zürich, I suggest you spend your Zürich days at the end. I personally think that Luzern is a better base, but Zürich isn't bad either. Not sure about Lichtenstein though.
As for skiing: the first week of Dec is early season, some resorts aren't even open yet. Zermatt will be open but is far from everywhere else, so day trips are not really an option. It is also "wasted" on beginners - the terrain is steep and there is nowhere to progress to after the learner slopes. Saint Moritz should also be open and might provide you with more varied options for sightseeing - but stay in Celerina rather than St Moritz, it is much cheaper, architecturally more pleasing, and still has direct access to the slopes.

This being said, you need more than 1-2 days of tuition to learn skiing/ boarding. 4 days or so is a more realistic timeframe to start enjoying yourselves... 6 days being ideal. And given the cost of tuition in Switzerland, you might want to have a look at Austria! Lech opens on 3rd December, for instance, and is easily reached by direct train from Zürich main station (plus a bus). It is not a cheap resort, but tuition will cost much less. It all depends on your priorities for the trip!

My first time on skis was at the ski town of Engelberg, near Luzern, when I was a university student in Germany. I wonder if this might be a good choice? Apparently it has 25 % beginner terrain and more then 50% intermediate, not as much expert terrain as Zermatt. I think a place like this might be a better choice for first-time skiers than pricy and glitzy Zermatt. Plus it is closer to Zurich—-1 hour 43 minutes by train, with a single change at Luzern. Spend your last 2 days in Luzern itself

https://www.powderhounds.com/Europe/Switzerland/Engelberg.aspx

https://www.engelberg.ch/en/winter/ski-snowboard/

Zermatt has lots of easy terrain on the glacier. Great views of the Matterhorn, which looks like you could reach out and touch it.