Hi Elizabeth and welcome to the forum!
So the first 1/2 day - actually it's LESS than 1/2 a day with a 1:00 arrival at Fiumicino - would be right off a transatlantic flight, and the 2nd would be the day you'd be heading off to Lucca for the Saturday-to-Saturday stay, right? Arg. Having to get your bearings in a hurry - you'll need to find your accommodation and unload your luggage before doing anything else - and in potentially jet-lagged condition would be stressful. That 2nd morning, while not as jet-lagged, would also be rushed as you'll need to be checking out in the late morning, getting to the station, finding your platform, etc. While I love Rome, we've also given it enough time TO love it. 99% of the people I know who didn't like it didn't give it enough time to get to know her. They were simply overwhelmed with the crowds and having to figure out where things are and how things work in a fat hurry. She's much easier to manage and enjoy in smaller bites! As is often said, Rome wasn't built in a day, and can't be seen in one! :O)
Same as Tim mentioned up about Florence, there is also the issue of the most-visited attractions needing advance, timed-entry reservations. If just a walk by the Colosseum would be enough for you, then that's perfectly fine. If you want to see the inside, then you're going to need to make the reservations a month or so before you go. Same with the Vatican museums...which you definitely do NOT want to try and see in jet-lagged condition.
Most of us don't recommend making timed-entry reservations for arrival day anyway as transport snags (cancelled flights; delayed flights; baggage delays) are not uncommon, and tickets are almost always non-refundable.
Florence: trying to see that one is a day would be frustrating, IMHO. The "Cradle of Italian Renaissance" has a wealth of treasures to offer IF you have interest in those treasures. Not a fan of Renaissance art or art in general? Then just a walkabout with a pop into a few churches (some have entry fees), and scenic spots would work for you. Some of the churches offer some really nice frescoes by Renaissance-era who's-whos that you can see for free; they can be considered art museums in-and-of themselves. Again, this would be a small-bite approach versus trying to deal with, say, the mighty Uffizi (advance timed-entry tickets are a must) or Pitti Palace on a compressed timeline. The Accademia will also require timed-entry tickets if trying to cover ground in a hurry; you don't have time to waste standing in a line.
Any chance one of the planned day trips from Lucca is to Florence, so you'd have more than one opportunity there? And any chance of arriving in Rome another day or two earlier and/or departing from Florence a day or two later than planned?
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