Help needed on Zurich layover plan

Your checked luggage should be automatically forwarded to your flight to Rome. You can check your carry-ons at one of the many lockers at Zurich Hbf. Passport control, in my experience is very quick, especially at that time of day. Trains go from the airport to Zurich Hbf continuously.

Assuming you already have boarding passes for your Rome flight, you will just have to collect your carry-ons out of the locker and get the next train to the airport. That only takes about 15 minutes, then go through security to your gate. So I would say leave Zurich Hbf about 2 hours before you scheduled departure and you should be fine.

I can't help you much as to things to do or where to get fondue, which is not really a Zurich specialty, Fondue is a Savoyard dish from French speaking western Switzerland, but I am sure it is available somewhere. There is a large tourist information office at the Hbf, they will be glad to give you some ideas. When I had time to kill, I visited the Landesmuseum, next to the Hbf, about Swiss history, from prehistoric to modern. Another time to the Kunsthaus, yes, an art museum. You could take a tram down by the lake. A day ticket bought at the airport to central Zurich will cover all your transport from/to the airport and within central Zurich.

Thanks! I’m confused about one thing.. could we store our carryons at the airport instead of taking them to the train station (assuming that is what Zurich Hbf is)?

Yes, there are lockers at the airport train station, just not sure exactly where they are. There are a LOT of lockers at the Hbf.

Hbf: abbreviation for Hauptbahnhof, German for main train station.

The main train station in Zurich has a huge shopping area (on many levels) and a large number of places to eat, to include at least one fondue place. The Landesmuseum is right next to the train station and an excellent history museum with very good displays of bronze age artifacts and reconstructions of typical Swiss interior design (think lots of wood and carvings). You can take a tram from the train station to the plaza just outside the Lindt chocolate factory at the north end of the lake. From there you have walking paths all the way around the lake, and a number of parks and attractions you could visit. If you don't take the tram you can reach the same point by walking through the old part of the city. It's a very easy hike, with minor elevation changes and narrow streets, but quite attractive.

Note that if you select a return ticket to the city on the ticket vending machine at the Airport you will in fact get a Day Pass, good for all public transit in the city, and between the airport and the city. So you are free to hop on and off trams at a will the whole day...

Zurich has many interesting places to see during your layover. I would suggest going online to search things to do in Zurich.

A fondue lunch would be possible, more so in the winter months. Caution: After a big fondue lunch, you may find yourself getting pretty tired from your overnight trip. Don't underestimate the exhaustion factor of flying to Europe unless both of you are heavy sleepers on the plane ride over to Zurich. Maybe a light lunch and digging into some Swiss chocolate might give you an energy boost. Enjoy.

Thanks Everyone! As always, I really appreciate the helpful people on this forum!!

I really like Rick's self-guided walking tour for Zürich which is in his Switzerland guide book. You can stop along the way for lunch, perhaps in the Niederdorf district. Check out Restaurant Swiss Chuchi for cheese fondue.

Getting through immigration in Zurich usually isn't a major ordeal, and at 6:10am you're going to be on one of the first planes to land, so maybe there aren't that many others at immigration yet.
Locker options at the airport can be found here: https://www.flughafen-zuerich.ch/en/passengers/shopping-and-enjoy/services/all-services/brand-baggage-room
Getting to Zurich HB / main station by train is quick and easy. Make sure you purchase a day pass for Zurich, saves you the hassle of having to think about a ticket if you want to get on a tram and when you go back to the airport.
Definitively have a stroll through the Niederdorf and the old town in general, it's very nice.
On Fondue, Zurich Tourism offers these suggestions: https://www.zuerich.com/en/eat-drink/typical-and-traditional/raclette-and-fondue (I would prefer raclette, but Fondue is equally delicious).
If the weather is nice, you may also consider heading to the lake (tram or maybe 20 min walk from the HB / main station) and take a mini-lake cruise. There should be day passes that cover this as well.
Don't underestimate time you'll need to get back to the airport and catch your flight - security check can sometimes take awfully long in Zurich; passport control is the usually somewhat faster.
There's also a tram (street car) line that runs from the city to the airport - a bit longer than the train, but maybe a fun alternative to have a ride through the city on a different route. You can check departure and travel times on sbb.ch; they also have an app that works well.