5 days 6 nights - flying in and out of Rome

Since you will need to be in Rome the night before your departure flight you should save all Rome nights for end of trip to eliminate a split stay/hotel change

6 nights is not long enough to add in so many day trips
Florence and Pisa are too far and too much as a day trip on same day

You will have Christmas and Monday closures to work around

You could head to Florence on arrival -3 nights with maybe a day trip to Pisa from Florence
Your arrival day is a wash due to jet lag anyway

Then back to Rome for final 3 nights

OR
stay in Rome the entire time and consider a day trip to Orvieto and/or Ostia Antica
There is certainly enough in Rome to fill your 5.5 days
Days are short in December and every time you change hotels you’ll lose at least a half day

You will need to book your entries to the major sites
Not sure what’s available at this late date

Search the forum for advice on guided tours and official ticket sites

If you arrive on the 22nd and depart on the 27th you have five nights, but only four full days on the ground. That's long enough for an introductory visit to Rome, but I wouldn't recommend side trips. That said, the train to Florence takes about 1.5 hours each way, so you could do that as a day trip if you want a quick look at another city that is actually worth multiple days.

Keep in mind also that many sights are likely to be closed on the 25th. You'll want to plan your sightseeing with this in mind. You'll need advance timed tickets for major sights like the Vatican Museum, Colosseum, and Borghese Gallery. There's another "Italy" thread currently discussing these options. If you go to Florence, you may also need advance timed tickets for sights like the Uffizi and Accademia.

There are tours that cover major sights in both these cities, including some requiring timed entry. You might look at Viator, a tour broker that doesn't actually operate tours, for an idea of what's available. If possible, try to book the actual tour(s) through the actual operator(s) as Viator takes a big commission from them.

Please do clarify what date you land in Rome and what date you depart, because it does seem as if you have only five nights in Italy rather than six. On a short trip like this, one night can make a difference in the advice you receive.

Tickets for the Vatican Museums (including the Sistine Chapel) are currently selling out, sometimes quite far in advance. The only date with ticket availability on the Vatican website for the period of your trip is December 23. Entry times available at the moment are 1 PM, 1:30 PM, 3 PM and 3:30 PM. There are also "Prime Experience" tickets available for December 23. They get you breakfast and an early entry to the Museums before the crowds pour in. If you want to see the Museums (likely to be very crowded and perhaps not the teenagers' top choice among Rome's attractions) at the most reasonable cost, I'd recommend grabbing those tickets ASAP. There may well be Vatican Museum tours available for other dates during your visit, but they will cost more.

https://tickets.museivaticani.va/home/fromtag/3/1671771600000/Biglietti-Musei

You might reasonably choose just to see St. Peter's Basilica, skipping the time-consuming Vatican Museums.

Make dinner reservations NOW! Suggest Armando al Pantheon for one of your nights. Another night if you like seafood, dine at La Rosetta. Terrific!
We were in Rome 12/27-1/3 a few years ago. Very crowded, mobbed, with Italian tourists. Beautiful Christmas lights strung across the streets. Enjoy Rome but have dining in place.

If your hotel is already paid for the 6 nights, I wouldn't spend nights elsewhere. You don't have a lot of time in Rome, which needs at least 3 days to cover the bare minimum.
You could do a day trip to Florence by high speed train (90 min each way), but even if you get there with the first train and return with the last train, you only have so many hours at your disposal, still enough to cover some attractions.
I wouldn't add Pisa to the mixt because that would add at least 90 min (each way) to the trip, and I don't think it's worth it for basically a poorly designed bell tower that was built on terrain that was too soft for that type of structure.
Another alternative day trip could be Pompeii (near Naples).
You can also do a day trip to Orvieto, which is smaller and closer to Rome.
Just make sure you have at least 3 full days for Rome, if you haven't been before.

Please do clarify what date you land in Rome and what date you depart,
because it does seem as if you have only five nights in Italy rather
than six

Echoing acraven's question; please clarify? Are you flying home on the 28th? What time is your arrival on the 22nd, and where are you flying from?

Also, can you share the name of your hotel? I'd like to run a quick check to make sure it's in central Rome.

Yes, you'll be working around Christmas, when most things are closed, and Monday when many of the state museums are closed. Expect the Italian cities to be busy, and if you're wanting to see the top sights (Vatican Museums, Colosseum, Uffizi in Florence, etc) order your tickets NOW.

I consider 4 nights/3.5 days the minimum to scratch the surface of Rome, and that's without Christmas closures. Personally, I'd spend all of your time there with the exception of maybe a day trip to Orvieto. There's a lot to see in Florence, and with daylight being short I wouldn't try to do that one AND Pisa in a day. If you DO decide to day trip to Florence, you'll need to do it on the 23rd, 24th or 27th, as some of the main sites, such as the Uffizi and Accademia, will be closed on Christmas and Monday the 26th.

I don't know as I'd plan a day trip out of Rome on Christmas Eve day (24th) but will defer to people who've done it; maybe it wouldn't be an issue..

Rome is a walkable city and there is so much to see where you do not need to take tours or hire guides. How old are the teenagers - and male or female or both? Have you asked them what want to do/see? Have you been to Rome before? Have the kids? Where is the hotel located?

One thing I did was do a walking/food tour of the Testaccio area of Rome which covered, food, history and a couple of ancient sites as well as great food. The same company (Eating Italy) does a similar tour of Trastevere and lots of other interesting foodie tours might appeal to the kids.

A trip out to the Roman city of Ostia Antica might be of interest. A complete City, the original port of Rome, you get a good idea of how working Romans lived. It’s an easy train journey from Rome.

Get Rick Steve’s’ Rome guide book:with that there is no need to pay for expensive tours.

Stay in Rome. Due to a flight cancellation, we only got 4 days in Rome (We were supposed to have 6) and it wasn't close to enough to see everything. It is the one city in our 5 week sabbatical that all of my kids (9, 13,15, 16) want to return to.

Stay in Rome, I have been there twice for a total of 11 days and still not seen everything that I wanted to see.

Correction to something in my previous post: I just checked and the b-ticket site (official ticketing site for many of Florence's museums) is selling tickets to the Accademia and Uffizi for Monday, Dec. 26th. This is unusual but likely because of the Sunday (Christmas Day) closure.

https://www.b-ticket.com/b-Ticket/uffizi/default_eng.aspx.html

25th [...] do a day trip to Pisa as it's open on 25th (kids interest)
26th take morning train to Florence [...] take the last train back to
Rome.

To go to Pisa, the fastest way is through Florence (Pisa is 50 miles west of Florence, about 50 to 80 min from Florence by train).

The train journey from Rome to Pisa is about 3 hours each way (with train change in Florence). The walk up fare is almost 120 euro per adult, or almost 80 euro per youngster under 30, round trip (but the adult can save over 30% if you buy a restricted non refundable non changeable ticket in advance).

So your plan above would entail two separate day trips basically to the same direction. It would probably make sense to combine the two into one trip, and maybe spend a night in Florence. With 3 people it might be cheaper as well, since the round trip to Florence alone would be at the very least 180 euro for 1 adult and 2 youngsters.

Agreeing with Roberto: now that you know that you can visit the Uffizi and Accademia on Monday (reserve your timed-entry tickets in advance!) it makes sense to combine Pisa and Florence into an overnight trip to economize high-speed rail ticket prices. As you'd need a hotel room for 3 on Christmas night, make a room reservation right away.

I read OPs post as if it’s a room/flight package booked with hotel in Rome
If that’s the case it makes more sense to just stay in Rome

Until OP returns to clarify it’s difficult to advise any further