The first thing that occurs to me is your flight plan. Rome is a very big international airport. Florence is a small airport. If you fly to Rome from Phoenix, you will have one plane change, either in the US or in Europe. If you fly from Florence, you will have to change planes in Europe, probably limiting your airline choices and likely adding one more plane change. I'd check out possible flights and prices first. Changing planes on international flights can have issues. If the change is in the US, you may have to change terminals from domestic to international, but your checked luggage will be transferred by the airline. On the way back, you'll have to go through passport control and customs after picking up checked bags before boarding the plane to Phoenix. Some airports are easier to go through than others. In any case it's best to fly with only cabin baggage. There are lots of tips on the Forum and from RS about how to pack light. You may find that flying round-trip Rome is better than flying open-jaw.
Secondly, as first-time travelers to Europe, and indeed, abroad, you should get a copy of Rick Steves Europe Through the Back Door and you and your wife should read it thoroughly, at least once. The RS Italy Guide is very helpful, especially for first-timers to Italy, both because of the very detailed how-to instructions and the self-guided tours of sights and city walks. There are good descriptions of the sights to help you choose what most appeals to you.
You should start your trip in Florence and end in Rome. If you land in Rome, you can take the train to Florence. There are some trains directly from the airport, though it's likely you'll have to change in Rome, which is not at all difficult. Rome is a very big, bustling, noisy city and its major sights are spread out. To be efficient, you'll need to use some public transportation. In Rome there are buses, trams, and the metro as well as taxis. It's easy to get lost - street names are hard to find and large intersections are confusing with streets veering in different directions. Florence is relatively easy. The train station is a few minutes' walk from most hotels. Most of the sights are walking distance from each other, the train, and your hotel. Starting in Florence allows you to adjust to all that's new and different - the signage, the money, getting oriented, etc. - and get over jetlag without the added stresses of Rome. I think you are wise to limit your trip to 2 cities and you've chosen them well.
There are frequent fast trains between Florence and Rome. It will take less than a half day to get from one hotel to the other. You can walk or taxi to the Florence train station (Firenze SMN, or Santa Maria Novella) and from the Rome station (Roma Termini) to your new hotel. If you go early in the day, your hotel will store your luggage until you return after your room is ready. If you go late in the day, your Florence hotel will store bags for you till you pick them up to go to the train.
Get used to walking a lot. You will go up and down a lot of stairs. Rome is hilly, so much more up and down walking and many more stairs.
My final advice is Less Is More. Slow down and enjoy everything you see and do. You will never be able to see and do it all. It's fine to have a detailed plan but assume things will go wrong, be flexible, and don't regret what you miss. It will be there next time you visit. Assume you'll be like most visitors to Italy - they never go only once.