Three nights in Rome

Three nights is so little time for Rome, and attractions are spread out quite a bit. It's going to be a challenge.

Make a list of your top priorities. Then set up a Google map and locate them. That way you can see what's near other things, and you can devote each day to a general area. Say one day for the Vatican and one day for the Colosseum and Roman Forum. And depending on your energy level, you can always put some things on the back burner, extra things to see each day if time allows.

Depending on your arrival and departure times, you may have some time available on arrival day and departure day, so you can pick your other highlights for those days based on proximity.

The main point is don't be random, or you'll spend too much time getting from one place to another.

hey hey vvece
when is this trip, where are you coming from when you arrive in rome? rome is always busy/crowded all year long with lines to see the attractions you're hoping to see. you will need to book reservations with time slots and dates, keep an eye on the sites when you can book with you large group. always curious about what "good travelers" means with young kids. if you wait for hours to get into these most popular activities are they contented waiting? do you plan any kid friendly activities with them
have you booked accomodations, how far from places, you can only go so fast as the slowest one without grumpiness. (window shopping, walking slow, hungry to eat. what does good restaurants mean. we use to walk down alleyways for local family owned uncrowded away from crowds and not waiting
you need to be realistic what you plan and see with this many people in you group. 3 days is sooooo short a time in rome and if arriving from USA/Canada you have jet lag and it'd real
hope all works out
aloha

So much missing information to give you good advice. Let us know:

the dates of your trip (weather, holidays, crowds all affect your itinerary)
if you are arriving from another city in Italy, in Europe or from North America (jetlag, lack of sleep can make your arrival day and first full day low energy)
roughly what time you get to Rome and what time (and how) you leave. Unless you are arriving early (and not after a red-eye flight) and leaving late in the day, you really only have 2 full days.
What's your budget? Are private guides/tours in or out? Taxis (you may need 2 for your group)?

Do you really want to spend hours in the Vatican museum?

Three nights in Rome, does this mean you’ll arrive in Rome after a flight from North America on day one? If so, don’t expect to do much sight-seeing after a night of no sleep.
Hiring a private tour guide might be your best bet. I don’t know if Rick Steves pocket guide suggests who to contact, but I would start there: https://store.ricksteves.com/shop/p/pocket-rome.

We are arriving at 7am from the US, so we will be jetlagged...we have traveled with these two grandchildren before and they were so easygoing - even with 5 hour flight delays in Iceland, and walking tours. For our first day, what can you recommend that is on the light/slow side since we'll likely be exhausted...maybe a family friendly cooking/pasta making class?

What time will you be allowed to check in?
Where do you plan to wait until check-in time?

You have such a short stay in Rome. I would not cut into that with a cooking class. Much of the joy in Rome is just wandering around, taking everything in. Do you really want to rush from "big attraction" to "big attraction??"
Do you plan to buy advance tickets for these attractions?

What is the location of the hotel, or apartment? If you are staying in an apartment, will you not have to spend some time stocking it after you arrive?

Have you visited Rome before?

Is this a pre-cruise visit?

You need to be walking not sitting or standing. Rick Steves has self-guided walking tours on his app and in his guidebook but they can be confusing. I would find a tour guide who will pick you up and let them take you on a Rick Steves walking tour. This way you’ll see it all.

One bit of advice, book your group early for the Vatican museum. Being in the first group of the day is heaven. The museum at 11:00 am is hell.

You won’t get into Rome ‘til mid to late morning, if your plane arrives at 7 am. Get to your hotel, where it will be too early to check-in but you can leave your luggage at Reception. Get some lunch. Walk around - perhaps the Forum and Coliseum. The Forum is a great way to feel like you’ve arrived in Rome on Day One.

Then you have two full days. Since the Vatican Museums and St Peters may be a bit of an ordeal for the kids (or adults) on the first full day, maybe save those for your last full day, which leaves your first full day on the casual side. Get one or more good guidebooks, either purchased or from the public library.

There are so many churches in Rome with amazing art … Bellini statues (Santa Maria del Popolo Church), Caravaggio paintings (San Luigi dei Francesi Church; Santa Maria del Popolo Church), and Michelangelo sculptures (San Pietro in Vincoli, or St Peter in Chains, has Michelangelo’s tomb of Pope Julius II with its statue of Moses). San Pietro in Vincoli is not far from the Forum and the Coliseum, so that one could be an arrival day add on to a walk through the Forum and the Coliseum.

Of course, St Peter’s Basilica has Michelangelo’s Pieta, but that will be part of your second full day when you go to the Vatican. Be ready to start the Vatican Museum first thing in the morning. Very first.

There’s the Pantheon, which fits into either of your full days. Ditto for the Trevi Fountain. The Capitoline Museum and the Villa Borghese may be too much for your 2-1/2 days in Rome. But look at guidebooks and decide for yourselves. The DK Eyewitness Guide is excellent to help you and your whole family visualize and plan, with photos and nicely illustrated walking tours.

Consider where to stay. Near Piazza Navona - probably most central? Trastevere? Spanish Steps?

And remember, gelato is always a hit with every generation of the family, day or night.

For our first day, what can you recommend that is on the light/slow
side since we'll likely be exhausted...maybe a family friendly
cooking/pasta making class?

If by "first day" you mean arrival day in Rome, I wouldn't pre-book anything at ALL lest you experience a flight delay or cancellation. Many pre-booked attractions and classes have no-refund policies. As well I wouldn't recommend a 'class' of any kind than requires close attention when jet-lagged, and no more than an outdoor walkabout in the fresh air to keep moving and stay awake until bedtime ROME time.

We two spent 2 nights in Rome in Aug 2019, and visited the 3 sites you mention plus more. Our trip report (part 4) describes how we used our time: https://community.ricksteves.com/travel-forum/trip-reports/italy-2019-sicily-sorrento-amalfi-pompeii-rome-orvieto-tuscany-siena-florence-cinque-terre-milan

Thank you Geoff B, great info.

Just note that Geoff and his group had been in Italy for about a week when they got to Rome.