Some questions about Florence planning

I’m wondering whether the climb is a must-do or if I’d be just as happy without it.

I wouldn't skip the endless views of Florence's orange and red tile roofs from the Duomo. I'd put it on the "must-do" list.

Finally, for those who love Florence: if you had to choose one or two favorite sites or experiences beyond what I’ve already planned

I'd also include the San Marco Museum and the Bargello.

For my taste, my first choice for an add-in is I would add the Medici Chapel to see a magnificent room plus the tomb art--the one chapel is basically one big room, so doesn't take long to see.

I hope that when you say Duomo complex, you are including the Duomo museum, which is well worth your time and increases your appreciation of the Duomo itself.

For something different that does not take a lot of time unless seriously interested in the science or the scientist, try the Galileo Museum which is near the Uffizi. It's a little break from both art and history.

For food, I'd visit the Mercato Centrale for outstanding food stalls, although not a relaxing destination for a sit-down dinner. (I only remember communal tables, but I wasn't looking for anything other than a place to sit either.) But very worth the visit for lunch or snacks.

One of the highlights of my 2024 trip was climbing the dome of the Duomo. I had purchased the Brunelleschi Pass. Being physically fit, I had no problem with the ascent. Along the way, I was behind a couple; the wife was extremely claustrophobic, but she continued upward with encouragement from her husband and me. She was thrilled to have conquered her fear. The reward of up-close views of the dome's spectacular frescoes, along with the stunning city views, is truly worth it if you do not have mobility issues.
There was nowhere I visited in Florence that disappointed me. My only regret was not having enough time, even though I managed to see a lot. Other personal favorites not mentioned in your post were Palazzo Vecchio, Orsanmichele, the Medici Chapel, and Santa Croce Basilica. And I must add the Cimitero delle Porte Sante at Basilica di San Minato. I spent hours there!

Florence is truly captivating! Enjoy every single moment of your trip!

Other sites apart from the really “big” ones:

The Bardini Gardens…quieter and prettier than the Boboli , IMHO.
You still get views of Florence from there.

The Bargello Museum….lots of sculpture.

Palazzo Davanzati…..a restored townhouse ….plenty of history there.

Museo Ospedale degli Innocenti…very moving …in a building in Piazza Annunziata where people used to leave unwanted babies hundreds of years ago.😢
Sad but fascinating, and up on the roof is a lovely open terrace and cafe.

Enjoy Florence, my favourite city in Italy….going back for the 14th time soon!

Fiesole….take the bus up….takes about 20 minutes…..lovely Roman ruins, amphitheatre….fairly quiet compared to Florence.

I enjoyed good Tuscan food in a relaxed atmosphere at Mattacena. Also loved the wild boar pasta at Osteria Toscanella. One of my favorite memories is walking up the hill to the viewpoint at Piazzale Michelangelo, then wandering through San Miniato al Monte cemetery. You can also take a bus up/down the hill, but I enjoyed walking through the neighborhood en route. (That may be part of the RS walking tour - can't remember.)

My favorite churches were Santa Croce (lots of famous people buried there) and Santa Maria Novella for the striped columns and general peaceful ambiance. Get a gelato at one of the places near Piazza Santa Maria Novella at sunset and then sit there to people watch - one of my favorite places to drink in the moment.

People on this site will be tired of hearing me say this, but the Museo San Marco is the rare opportunity to see art in its original setting. I also love the Galileo Museum. I enjoyed the Medici Palace, but it wouldn't be a must-see for me.

September is likely to be very hot. There will be crowds all around and inside the most popular and famous sites. I just want you to take that into consideration when you plan --- you may not get to all the things on your list, or feel like, say, walking around Boboli.

Florence has so, so, so much to see it makes planning difficult. I almost hate to add to your list....

Thinking about history rather than art, you might enjoy finding (or just noticing) the flood markers from the 1966 and previous floods high on the walls --- some on streets, some in the courtyard of Santa Croce, one is inside a shop. Before you go, read "Diary of Florence in Flood" or at least the first several chapters. It was written by someone who was right there, staying in a guesthouse next to the Arno, watching the waters rise and seeing the destruction and the clean-up afterward. Her book has been reissued recently under the less good title "Florence: Ordeal by Water." You can see videos of the 1966 flood on Youtube.

Museo Archeologico Nazionale --- won't be crowded and is excellent.

La Specola --- I expected to dislike this natural history museum but loved it. So much weird, weird stuff there.

Try to see the Benozzo Gozzoli chapel in the Palazzo Medici Riccardi --- a tiny room where you could easily spend an hour (but you don’t have to, unlike most of the other churches, museums, & art galleries) looking at all the details of clothing, portraits of the Medici family, animals, horses, flowers, etc. I always skip the rest of the palace. The paintings covering all the chapel walls seem to be saying “We Medici are like the three wise men bringing gifts, so it’s OK that we are filthy rich.” The room fills with people and empties every few minutes.

Look for Clet Abraham’s clever altered street signs. https://www.cbsnews.com/news/trafficking-in-art-clet-abraham-street-signs-in-florence/

I've never been to the Pitti palace and never wanted to during a total of about 80 days spent in Florence, despite attending many art history lectures about it. But I try not to discourage people from going someplace I haven't actually been to myself. I was not excited by the Boboli gardens (and I am a very serious gardener!). Very skippable, in my opinion, when you only have four nights and the weather is hot. I always worry that the Uffizi/Pitti/Boboli pass makes people feel like they "should" do all three.

The view from San Miniato al Monte is wonderful and free!

For meals, we liked Trattoria dell’Orto, Trattoria Sabatino (no reservations but the line outside is fun as is the lively old-time osteria style of the place), Alla Vecchia Bettola, the deli Gastronomia il Giglio, & pizza at Fermento 1889. All in San Frediano.

Outside of San Frediano we enjoyed Acquacotta. And eating at the Sant'Ambrogio market at Trattoria da Rocco --- this also has its own style and you sit in a booth with random other people.

Florence is my favorite city on earth and I hope you love it and will return to it!

The San Croce church is amazing, it is the burial place of many of Tuscany's famous people.

Florentine illustrious figures
The tradition of using the church to honour the great and good was revived by Cosimo de' Medici when he commissioned a tomb for Michelangelo in 1564, thus laying the groundwork for the Grand Duchy of Tuscany's cultural policy based on using funerary monuments in Santa Croce to celebrate the genius and talent responsible for Florence's greatness. The tombs and memorials to illustrious figures include that of Galileo Galilei, who died in 1642. His funerary monument, commissioned by his pupil Vincenzo Viviani, had a complex gestational history that was only resolved in 1737; and indeed the same is true of Niccolò Machiavelli, who died in 1527 but who was not celebrated with a monumental tomb until 1787.

If you're more into history than art, perhaps reconsider the Pitti Palace. It has some amazing art (Raphael and Artemiai Gentileschi, among others), but it is a rather disorganized museum. Instead of trying to wander through, decide which artists' works you want to see the most, and try to get to the rooms where they are. Or maybe you'll decide the hassle isn't worth it to you, based on your available time and interests; that's ok too!

Santa Croce is magnificent, and depending on your feelings for seeing where dead significant people's tombs are, you'll get to do that too.

Santo Spirito, on the same side of the river as the Pitti Palace and Boboli Gardens, is a beautiful church.

If you like Michelangelo, then definitely plan to visit the Medici Chapel, to see his sculptures there. To me, it's more meaningful to see art in situ, where it was meant to be placed or seen