Recommendations 13 nights in Italy - train

That's easy (at least for me ). 3 nights in Venice. 4 nights in Florence, with 1-2 day trips to Sienna or Lucca/Pisa. The rest in Rome, with maybe one day trip (Ostia Antica or Tivoli.)

I basically agree with CJean but I’d add a day to Venice to allow a day trip to Padova. An interesting city with the “don’t miss” Scrovegni Chapel. Frequent trains take a half hour from Venice to Padua.

CJean's plan is exactly what I would do

And another vote for CJean's plan. 👍
If you do have 1 more night (not sure how your 13 is counted), then add it to Venice or Florence.

thanks so much for your advice! we appreciate the day trip ideas and are looking in to those ideas, too. 13 nights does not include the night of travel time from US to Italy. We are due to arrive in Venice on a Saturday morning at 11:00 a.m. (Saturday night counts as our first night) After reading other forum advice, it seems as though it will make sense to begin our activities/tours in Venice on Sunday after recovering a bit from jet lag. Please correct me if I'm wrong about that.

it will make sense to begin our activities/tours in Venice on Sunday
after recovering a bit from jet lag.

That's what I would do. Jet lag can affect everyone differently but it's usually a wise idea not to plan anything - like museums - that demand close attention or hours indoors. Walking about in the fresh air and generally getting acclimated to new surroundings helps one stay awake until bedtime. Some folks can take a short nap but that doesn't work for us: we'd wake up feeling worse than immediately hitting the streets. A nice cup of good Italian coffee helps, too! That's one of the first things we do after dumping the bags.

It's also best not to pre-book any organized tours for arrival day, just in case you experience a significant flight delay or cancellation.

Thank you for that advice, Kathy!

You're welcome!

A word about Sunday, your day of arrival. I don't usually like to micromanage my meal-planning, but I would do due diligence in research as to what restaurants near your hotel in Venice are open on Sunday. Last year in Rome, I was tipped off to this phenomenon in advance and booked Emma Pizzeria before we left the US. Good thing--we were jetlagged upon arrival on a February Sunday (right about this time of year now that I think about it), not horribly but still we needed a good meal, and this one was great. Spaghetti alla Carbonara...ah, transcendent! Then after a nice walk and some gelato we slept like babies and were fresh as daisies Monday morning.

Enjoy your planning!

That's a great idea, Jay. Thanks for mentioning that! And, thanks, also, for the tip on the restaurant in Rome - I love good food memories! My last transcendent food experience was eating Barbecue Shrimp at Mr. B's Bistro in New Orleans!