Hi,
Have you considered looking at trip itineraries of major tour companies going to Italy?
They are logistically planned out and would help you in your own travels.
Also, there are local tour companies who can give you a half day or a full day tour of Rome and Florence.
https://www.walksofitaly.com/
https://theromanguy.com/tours/italy/rome
If you will be staying in Milan for a day or so, the major attractions are the Duomo, the Last Supper, and a visit to the galleria.
Rome is my favorite. Around every corner is an architectural surprise. The city is an outside walking museum.
If you are planning to go to the Vatican on your own, here is the ticketing website:
https://m.museivaticani.va/content/museivaticani-mobile/en.html
At St. Peter's Basilica, you do not need tickets whether you visit the Vatican or just St. Peter's Basilica alone.
If you want to climb the dome or take a Scavi tour, there is a fee.
If you are interested in visiting Castel Sant'Angelo, it is a short distance from Vatican City. There is a fee for tickets.
The Pantheon is a beautiful church. No entry fee is required or at least the times I was there. Not really a security check but there are guards outside just observing everyone.
Inside The Pantheon is really beautiful. If there is a service going on, visitors will not be allowed to "tour in" unless they are participating in the service. The surrounding plaza - Piazza della Rotonda is quite charming. Surrounding it are little shops and cafes.
Many churches, small and large, are free and usually open to visitors. The churches can be a museum of sorts.
At the Vatican, St. Peter's, The Colosseum, and The Borghese Gallery, there are bag restrictions and security checks.
A day bag/small backpack of about 14" is usually allowed at these attractions after the security check.
One exception is The Borghese Gallery - they do not allow larger bags at all. If you visit the BG, just wear a very small crossbody under your coat for valuables.
Restaurants will be open on Christmas but reservations are strongly encouraged or needed in some restaurants.
If you want a quick espresso, panino, or gelato, there are many cafes everywhere. Be mindful if you want to sit down, even outside, the eatery will or may charge extra for sitting.
There are or were two grocery stores under Roma Termini. There is also a food court. At the market, you can grab a panino, water, etc.
Besides the majestic Trevi Fountain, the Spanish Steps and the Trinità dei Monti Church, the beautiful Piazza Navona, a visit to Campo de Fiori is a delight. During the day is the street market, pricing can be a little high for obvious reasons. By late afternoon, the tents are dismantled and the nightlife begins - restaurants around the Piazza will be more expensive. If you want to eat in that area, stroll the little side streets for restaurants with decent pricing and good food.
My trips to Rome have been in December.
Average weather will be in the 40 - to 50 degree range during the day and in the 30's at night.
One year, the day temps reached 70° for a day or two.
Another year, day temps barely hovered at 40° with strong winds for a couple of days.
Layering with fleece or sweaters under a coat or jacket is your best bet. A hat, a scarf, and gloves are what I would recommend too.
There may be rainfall as I experienced once or twice. So that will obviously make it seem colder. Most people carried umbrellas. If you do not want to take an umbrella with you; no worries. Vendors magically appear to sell umbrellas for 5 Euro at the first drop of rain.
https://www.romewise.com/ < Check out this website for ideas, suggestions, etc. The author is "boots on the ground" - married to a Roman.
It is always best to double-check on direct websites of any attraction or restaurant for policies, changes, timing, and pricing.
Have fun.