Hi arezou_bk,
Perhaps you can use this list as a guide:
Vatican City - Be in line very, very early - Unless you schedule a guided tour. The gift shop in the Vatican Piazza have "local guided tours" available or if you can get lucky, I think the Vatican has their own too. Check the website for accuracy.
There is a security check, no "big" backpacks or bags or large umbrellas. Don't bother with the coat room - you would waste time. Head right to Saint Peter's entrance. If you are not with a guide, you may have to wait in line for His church.
Saint Peter's Basilica - Free to enter. Much more doable compared to Vatican. More breathing space, more time to view altars, statues, paintings, absorb the whole aura.
Stay away from those "offering skip the lines" tix.
Castel Sant'Angelo - "Bridge Of Angels" - A lovely walk down on the bridge. (No reservations)
Piazza del Popolo - "The People's Square" - Large, urban. (No reservations)
Spanish Steps - "Piazza di Spagna" - A "great climb to the top of the stairs" - Historic Church at top - Trinità dei Monti (No reservations)
Trevi Fountain - Gorgeous. Beautiful baroque fountains. Water pooling is very mesmerizing. A little "touristy" for food/gelato pricing. (No reservations)
The Pantheon - A beautiful (active) Church (there may be a small euro fee to enter now - it was talked about a few months ago. Maybe? €2 or €3 each. Be prepared just in case. (No reservations)
Piazza Navona - Home to "Fountain of the Four Rivers" (No reservations)
The Colosseum - You "might" get lucky be able to purchase tickets to enter (from coopculture.it) (Need tickets to enter for Colosseum, Roman Forum and Palantine)
Campo de' Fiori - "Open Market" in the mornings. Area & little back streets have cute shops/eateries all around. (No reservations)
If you can, click on each Roman attraction to get a feel of what you and your family may want to visit:
https://www.skylinewebcams.com/en/webcam/italia.html
Rome is a walking city and a walking museum as I always say. The city buses (if you tire and want to ride one, costs €1.50 each for 100-minute ride (must be validated on the bus - machine is near driver) Tickets can be bought at most gift shops or tabacchi shops. A "bus/metro pass" may not always be worth it in terms of cost vs usage. If you don't use your bus ticket, it won't expire.
With each attraction or interest, it may be best for you to search each one and see what it/they have to offer you and your family in terms of time, interest and valuable memories. Have fun!