How far in advance to buy tickets for Colosseum in Rome in March?

Hi Andrea - welcome to the forums!
In truth, March is not really off season at all for Rome and Florence anymore. You'd want to order tickets for the most-visited attractions as soon as they go on sale. This is especially important if you want tickets for a guided tour of the Colosseum purchased through their own website as they are snapped up very, very quickly in advance. You also have the disadvantage of traveling with multiple people (now many in total, and how young the children?) so need to land more than just one or two available tickets.

You will stand a much better chance of landing Colosseum tickets if you can forgo tickets which include the underground: they are the most in-demand ticket in Rome and have become very difficult to get. I've never considered that part a "must do" anyway as you can see down into it from uncovered areas above. Look at purchasing the ENGLISH DIDACTIC VISIT FOR INDIVIDUALS AND FULL EXPERIENCE TICKET tour on this site:

https://ecm.coopculture.it/index.php?option=com_snapp&view=products&catalogid=91E202AB-FCB0-F115-0C11-0174490A6193&snappTemplate=template3&lang=en

They go on sale 30 days in advance. There are other tours you can purchase from independent tour companies if you can't get hands on Coopculture's; they will cost more. As well, you might be able to purchase tours from the ticket kiosks at the site but I wouldn't count on. it, it could involve standing in a long queue, and you have to have your entire party with you. You must have your passports with you, and must have them on you for entry to Colosseum so they can check your identities against those on the tickets. You should always carry them anyway. :O)

https://colosseo.it/en/opening-times-and-tickets/
(This is also an "official" website; it only loads during hours the park attractions are open)

Vatican Museums: their tours go on sale 60 days in advance; they are currently available for purchase through April 3. Buy as soon as they are available for your desired date. This particular attraction may be tough on your very young people given the size of the visiting mobs: unless you MUST see the Sistine, I'd personally skip it. No tickets or tour needed for St Peter's but the queue to get inside can be very long...although it moves along pretty quickly. Try to do that one first thing in the AM or later in the afternoon.

https://tickets.museivaticani.va/home

Florence: https://www.b-ticket.com/b-Ticket/uffizi/default_eng.aspx.html
This is the official ticketing site for the state museums in Florence: these tickets are for entry, and not for tours. IMHO, you do not need a tour for the Accademia ("David" is there), and consider skipping the mighty Uffizi with very young children unless really into art as it's a very busy place. There are other less-mobbed attractions in Florence to admire some of the wonderful Renaissance sculpture and painting, and some will cost little or nothing at all.

You might consider a general walking tour of both Florence and Rome? Length of those you would look at booking may depend on the ages of the children. Other posters will have some recommendations for those they've enjoyed.

One more....
This is the info for advance tickets to the Pantheon in Rome. They do sell tickets there but the line could be very long. The website for advance tickets - which I've not yet used myself - is reported to be sort of wonky so better to buy in advance at home with good internet than in Rome. You may still run into some issues so post if you do for help from others who had the same issues.
https://direzionemuseiroma.cultura.gov.it/en/pantheon/

May be a useful thread:
https://community.ricksteves.com/travel-forum/italy/tickets-to-the-pantheon

Also be aware that ticket purchases for attractions may be declined if your credit cards aren't set up for the Verified-by-Visa 2nd layer anti-fraud protocol, Mastercard equivalent or that for other cards you are using. Those two are the most commonly accepted in Italy, though.

Last year I spent ten days in Rome around the middle of February and another ten days around the middle of March. I would agree it was a hopping place; you'll be surprised at what you encounter. A lot of the people walking around were undoubtedly locals, but every time I walked past the Fountain of Trevi (at least three times) the small surrounding area was mobbed to the point it was difficult to navigate on foot. Parts of the Vatican Museums--most definitely including the Sistine Chapel--were very crowded, though I know they are worse at other times of the year.

I don't know whether either of these sights in Rome will be of interest to you since you'll have two young children along, but of course the adults might want to split up occasionally. For both of these places you need to already have tickets in hand when you arrive:

The Borghese Gallery is a much, much more pleasant place to see great art than the Vatican Museums, because it is much less crowded. It regularly sells out all its 2-hour entry slots, so you will be shocked to see how many other folks are there at the same time as you, but believe me, the Vatican Museums are much, much worse.

https://www.tosc.it/en/artist/galleria-borghese/galleria-borghese-2253937/

San Clemente, a very interesting tri-level church, requires visitors to buy tickets online; they cannot be bought in person at the church. I don't think San Clemente regularly sells out far in advance, but you don't want to be standing outside the church, trying to get a cellular signal so you can buy tickets on the website.

https://basilicasanclemente.com/eng/booking/

With respect to the weather: I was very lucky in that I don't think I had a single day when the high temperature failed to reach the average for the month as shown in the Wikipedia entry for Rome. However, the daily high temperatures were not reached until close to noon; it was pretty chilly at 9 AM or 10 AM. And the temperature dropped late in the afternoon, so much of the day was coolish. Take layers and a waterproof jacket or coat and expect to be carrying one of those layers during the warmer part of the day. Folks not wanting to make a fashion statement by tying a sweater or jacket around their waists could use a small backpack (not for wallets!) or lightweight tote bag to stow them when they're not being worn.

As far as advance tix for Vatican museums go, I was able to purchase an official guided tour for Late June. This was for guided tour only. Non-guided were not yet available.

We were able to purchase tickets to San Clemente to the lower levels for 12 euros in person (on line was 10 euros). We were there this week. Super interesting.

Eef: How did you do that? Their website says you must buy on-line. Do you know if it was because it’s off season?

Lizcat, we just showed up. It wasn’t busy. We were given the option to buy on line but it was easier (for us) to pay cash and break a larger bill in the constant quest for smaller change. It is off season, although some places are still busy.

That sounds like a policy change. In the first half of March last year, San Clemente wasn't selling tickets on site.

As far as advance tix for Vatican museums go, I was able to purchase
an official guided tour for Late June.

Lizcat, what website did you use to book? I'm looking again at the "Guided Tours for Individuals - Vatican Museums and Sistine Chapel" ticketing section on the Vatican's own website and anything beyond April 5th - more than 60 days out - shows as "Not Bookable", meaning those tickets haven't gone on sale yet.

https://tickets.museivaticani.va/home/fromtag/2/1712379600000/VG-Musei

Kathy: I bought the tix on the official site museivaticani.va on 1/19/24 for English guided tour on June 25. Received booking voucher and credit card was charged.

I just checked the website in Italian ( can’t get English on the iPad) and I agree I don’t see it as bookable now. Don’t have access to desktop computer right now. Site works better on it.

The Roman booking sites have been frustrating.

Hi - I wanted to offer a quick word regarding the guided tour in the Vatican Museum. We will forever be sad we rushed desperately to keep up to our tour guide as the head phones didn't work well with either the crowd or the distance apart or just inside the building. By hurrying to keep up, we missed the vast majority of what was there. The head phones crackled so badly, we couldn't hear him even if close by. Perhaps he needed to hurry us through for the next groups - all fair/possible.

I realize he had a lot to offer but we could have/should have stayed on the outside of the travelling group/mass and let them hurry past us so we could see the tapestries, etc. The crowd was a packed elevator (in the amount of people) in motion. Small children ( or those of us that are shorter) would best see and move on the outside edges. Perhaps learn a bit more of what you will see as you move/shuffle toward the Sistine Chapel and not be dependant of the audio and find a way to not be pushed so much along.

Just my two cents - it was an amazing tour up to that point and we did appreciate the September crowd may be the better compared to summer season masses but it is something to keep in mind. Next time, I will not take the audio tour and hug the wall much more so I can see the tapestries, really look in each room, the walls, the art.... It is an extraordinary place and a forever memory to see a completely empty St Peter's Square at 7:30 am. But in my lack of information, I missed part that I really did want to see. I will certainly go back - but a little smarter :).