Starting in Switzerland and ending in Rome

Don't worry about connection times in Switzerland, they are always short by design, since the Swiss figure you don't like to waste time hanging around train stations. You can always take earlier trains if you like. Changing in Milan is likely to be and least 20 minutes. Florence to Rome is direct.

3 stops in 9 days is reasonable, Lauterbrunnen to Florence will take at least 6 hours.

we are trying to see as much as possible

What you will see are the interiors of train carriages. I received good advice before the first venture to Italy last Fall - it was slow down and take time to experiece a single region at a time.

We spent nine nights in Rome - could have stayed longer. I'd suggest terminating your trip in Florence - add days there - add Siena - or if you feel the need to see more, consider adding Venice. Save Rome and points south for the next trip.

This looks like an itinerary for Italy. 🇮🇹
I would focus on Florence, Tuscany and Rome and look into adding Siena, Lucca and Orvieto.
Save the Berner Oberland for a future trip.

When are you planning to travel? I ask because there are planned construction projects and other track closures that may complicated travel between Switzerland and Italy at times this summer.

But in any caseI would say that is too much train time for 9:days. If you want to start in Zurich and finish in Italy, consider a more compact trip, something like Zurich —- Luzern—- Lake Como—- Milan ( for your flight out).

I understand the desire to see as much as possible when you haven’t been to Europe before, but this itinerary means you’re basically going to have 4.5 days of vacation and the rest of the time is going to be spent in transport. I’d strongly recommend narrowing down your stops to give you more time to actually see things. You basically have 7 days. With that amount of time, pick Switzerland or pick Italy. As said above, it’s hard to give exact recommendations when we don’t know what time of year you’re going. And if you already have a flight booked into Zurich and out of Rome that obviously limits how much you can change this, but I’ll still recommend 2 bases instead of 3, regardless.

I actually think you have 6 full days. Florence to Rome is only 1 1/2 hours on the train. Yes you have to get to your new lodgings and drop your bags, but then you have the rest of the day in Rome.

Everybody will always tell you to take more time, and they should. The Berner Oberland, Florence/Tuscany, and Rome each deserve a week. But many of us don't have the time or the money for a 3 week trip. So my advice is: try it, if you like it, you can do it again at a slower pace. You will learn from this. Its a good start.