I had your Italian adventure in March of this year. My DD and I spent 9 days together on this adventure.
#1. Watch Medici on Netflix before you go to Florence. There are Medici references all around the city, and you'll have a better knowledge of the family and city. (Assuming you are like I was with no prior knowledge of either.)
#2. Everyone knows about the Duomo, but make sure to go through the Basilica di San Lorenzo. It was consecrated in 393 and many of the Medici's and the Medici popes are buried there. It looks very plain and old from the outside, though the inside is gorgeous! Donatello is also buried there. He also made the lectionaries in the church.
#3. Best meal that we had in Florence was at Corte del Pazzi Trattoria. Unfortunately, we went on a Friday during lent, so we couldn't eat meat. They had pig cheeks on the menu that I would have had. Instead we had artichoke pasta dishes and as appetizers (prima piatti) we had black cabbage flan and fennel au grautin that were amazing!
#4. I found an apartment through Expedia that I rented called First of Florence. It would have slept 5 people with the pullout sofa. It was 2 blocks from the Duomo. Perfect location and very nice accommodations for somewhere between $112 and $120 per night. The staff supplied milk, Nespresso pods and sweet rolls and a loaf of bread for breakfast.
We had 4 full days in Florence and took the train to Venice. We were business class because I booked it. My DD told me we could save a little by booking the train from Venice to Rome online. So I told her that we wanted Executive or Business class. She booked it without me being present. She booked coach class, which turned out to be just fine. Train travel isn't like being herded like cattle onto a plane.
Make sure to research if Venice floods during June. I took rubber shoes with me and we bought my DD plastic boots for over her shoes because the streets flooded each night and while it was raining one day, during the day, too, to some extent. They put up risers for those who don't want to get wet, even going into St. Mark's Basilica. We spent 2 days in Venice but didn't get to do half of the things we would have liked to do. We stayed at a 4-star, small hotel, Ad Place. It was lovely and ~$120 per night. Breakfast included.
The pope has an audience every Wednesday. You can get tickets for free on Tuesday from the Swiss Guard. If the pope is in Rome that day and you get tickets, don't run to the front like everyone else. Go to a barricade. The pope will go right by you in the pope-mobile. I have a photo of him from about 5 feet away. He seems to be looking right at me. That same day, we toured the Vatican museum and St. Peter's Basilica. That night, we went to Trevi Fountain and the Spanish Steps. There was a prayer service going on in the church at the top of the steps. The folks inside were very welcoming and had us write our prayers on paper and walked us to the altar to put them into a basket and pray before the exposed sacrament. We prayed quietly and then our guide, Ludwig, prayed over both of us. It felt like it was meant for us to be there that night.
Next day, we did the Coliseum and Roman Forum tour. Fascinating!
We spent 3 days in Rome (for free) at the Hilton Aleph Hotel. This was a 5-star hotel for which I used points. 83,000 points per day when we were there and worth every one of them. It was 2 stops away from the Vatican on the subway. I amassed my Hilton points through my Hilton Amex. You have time to do the same, if you wish.
So, that was 9-days.
I haven't been to Greece before. How long are you planning to travel in this trip? It may be worth it to break it into a trip this year and one next year if you don't have enough time to do both countries justice.
