Recommendation for Zurich to Lucerne with children (and then on to Italy)

Luzern is a good choice. You can catch any of the frequent trains to Luzern from the train station right at the Zurich airport.

There are two trains an hour from Zurich airport to Luzern. The one at 25 minutes past the hour requires a change of trains at the main Zurich station. The one leaving the airport at 49 minutes after the hour goes direct with no changes. That one would be easier for your family.

As for the train from Luzern to Milan being "a beautiful ride", you may be disappointed. The main route now goes through the very long Gotthard Basistunnel, under the Alps. You will be riding in a dark tunnel rather than viewing the mountain scenery.

There is a way to avoid this by taking regional trains over the older Gotthard Pass route, but it would involve careful planning and several train changes. I will see if I can find that information for you.

You have a great idea of staying in Lucerne/Luzern for the first 2 nights for your young family to recover from jet lag and enjoy the lovely charming old town and lake. The train you should take from the Zurich airport to Luzern, the one where you (and your luggage) don't have to change trains, leaves the airport station every 47 minutes after each hour.

For a truly awesome gorgeous ride through Switzerland to Milan, instead of going the less scenic quicker way through the Gotthard Tunnel (whether the new one or the old one), take the route that goes via the Brunig Pass and via Interlaken Ost. You'll pass beautiful mountain lakes that way. Looking it up on my SBB app (www.sbb.ch/en), I see the difference is that the trip takes about 3 1/2 hours if you go through the new faster Gotthard Base Tunnel, vs. the trip through the Brunig and Interlaken route, which takes either 5 1/2 or 6 1/2 or 7 or 7 1/2 hours, depending on what time you go--they leave Luzern at 5 minutes after each hour. (For this scenic route, you would have to change trains 2 or 3 times--the fewest changes are with the 11:05 train.)

Swiss transport museum is in Lucerne. If the weather is uncooperative it’s a good option for the kids. The transit pass provided by your hotel will allow you to ride the bus to and from.

The transportation museum in Lucern also shows a great IMAX movie

Yes on train to Luzern from Zurich airport. Airport train station is below the airport terminal with staffed ticket counter. You don't leave the terminal complex to get to the train platform. As noted by others one train an hour goes directly to Luzern with out transfers, the other connects at the Zurich train station.

We arrived late afternoon as opposed to your early morning. We stayed at the Waldstaetterhof hotel, a half block from the train station simplifying arrival in Luzern. There are several other hotels within a couple of blocks from the station. I would expect that you could leave luggage for the morning. As an alternative SBB has a luggage service where you pick up the luggage at the train station counter: https://www.sbb.ch/en/station-services/services/luggage-and-flight-luggage.html

We did the same in June. As others say take the direct train from Zurich Airport to Luzern. We stayed in hotel Monopol right across the station. Short walking distance to the lake and center. The train to Milan goes from Zurich. Instead of going all the way there go to Arth-Goldau where you change trains and board the train to Milan. You will go through the longest railroad tunnel in the world. It's 57 kilometers and goes under the Gotthard Pass. Unfortunately that would deprive you of the best scenery. But scenery is still very good even without Gotthard Pass. In Milan you change trains for Florence. We went by superfast train (300 km/h). Only one stop - Bologna - and you are in Florence.