Italy May 2018

From Florence, Pisa is an easy trip, but you can see it quickly. Lucca is only about a half hour farther and much more interesting.
From Rome, Orvieto is just over an hour away, a favorite of many people here. Or Ostia Antica, if you are interested in ruins.

Your trip looks lovely. In Florence, try to get up to Piazzale Michelangelo to see the sunset. The view is wonderful. Or go to vespers at San Miniato church to hear the Gregorian chants, then watch the sunset from the front steps of the church. (My legs weren't up for the climb, so we took the bus). We were there for 4 nights and didn't come close to seeing everything we wanted to, so you may not feel the need for a day trip. But...

Lucca is often recommended on this site as a day trip from Florence, either on its own, or combined with Pisa.

From Rome, Ostia Antica or Tivoli are day trips that are close by.

I'd definitely buy advance train tickets for any of the routes using the high speed (freccia) trains. While they commit you to a specific train at a specific date and time, they can also save quite a bit of money. Regionale trains cost the same whether bought in advance or on the day, so there's no advantage to pre purchase.

Jim,

Lastly, am I better to purchase advance tickets for my three trips from M-V, V-F and F-R, or buy them on the fly? Thanks.

I’d suggest booking these tickets in advance only if you want to save money. Keep in mind that discounted tickets
(Supra economy and Economy) tickets are nonrefundable and unchangeable, so your dates must be firm.

In terms of taking day trips from Florence... If you’ve already visited Florence in the past, you may consider staying in Lucca, a charming small city surrounded by a great medieval wall wide enough to walk or ride a bike on. If you like classical music you can attend a Puccini concert held in a church and the cost is only €20! If you stay in Lucca you can easily travel to Pisa via train.

A day trip from Rome? I’d suggest visiting beautiful Orvieto!

Have a wonderful trip!

Edited to add...
I was writing prior to seeing the previous replies (using my phone:-)) but it sounds like we all agree on visiting Lucca and Orvieto!
Enjoy!

I love Lucca...Florence not so much. But not the fault of Florence...just too crowded for my taste. But I loved Pistoia which is less than a hour from Florence. I am returning to Lucca and also Orvieto this upcoming May trip. I will spend the first part of my trip in the Puglia region.
I will be flying into Pisa from Brindisi airport (from Lecce in the south) but will only take the train to Lucca (about 20 minutes) .
From Lucca I will take day trip to Poretta Terme where I am fairly certain I will be the only tourist there. Then return to Lucca , train to Certaldo because it looks interesting and again not on the typical tourist route. On to Orvieto 4 nights one of my favorite places and back to Rome for flight home. All of plan "flows" very well using public transport with the exception of the Ryanair flight from Lecce. Saves a lot of my time and cheaper than a train for that distance.
Have you booked your accommodations yet?

Re train tickets..just bought a first class train ticket from Rome to Bari for 35 euros. I am sure this is a savings over waiting until I arrive.

First, definitely buy tickets ASAP for savings. I haven't been to Lucca, but others say it's a good fit with Pisa. Leave early, stop in Pisa for 2-3 hours to see the Leaning Tower (buy a ticket in advance if you want to climb it), the church, the beautiful baptistry and the grounds, then take the train to Lucca for the rest of the day. In Milan go up to the roof of the Duomo to walk among the spires. I also enjoyed the La Scala tour and a visit to the Sforza Castle. I would not fret about not visiting small towns - it's all good. To get away from the crowds in Florence, you can take the bus up to Fiesole or cross the Ponte Vecchio and wander in the Oltrano (literally beyond the Arno). Orvieto is a good day trip from Rome. Ostia Antica is a 1/2 day, go in the morning since there's not much shade as you wander a 2000 year old Roman seaport.

With only 2.5 - 3 days in Florence, I personally wouldn't do a day trip, but I admit that I am biased (I studied abroad in Florence and it's my favorite city on earth). I love small-town Italy, but if you're the museum type (like I am), then the big cities are the place to be. I agree that Piazzale Michelangelo is a must-do, and wandering the Oltrarno (where I lived when I was in Florence) is a great break from the tourist hoards. Orvieto is lovely and would make a good day trip from Rome.

My thanks to all of you for the info and suggestions. Since I will only have two full days in Florence plus 1/2 day when I arrive I think I will stay in Florence. Since I have more time in Rome I'll do a small town day trip there. I saw a video today on You Tube shot from the Piazza Michelangelo looking toward the Duomo and just added that to the Must Do list, as is that picture. I always wondered where Rick gets that picture on his TV show and now I know. Anyone know where to catch a bus to go there, as someone suggested? I'm staying in a small one BR apartment between the train station and The Doumo. Thanks.

Jim