We are like you, we like to walk cities and see what we see. My best advice is to go buy the Rick Steves Guide to Rome - best simple breakdown of what to see, what's not I've read. He has a series of simple walking tours that are fantastic. Once I read the tour, I throw the book in my bag just in case and walk the general route. It's nice to have a bit of background of what you are seeing and helpful to explain to your son.
Get a Roma Pass - there's a special short line to get into the Colliseum, off to the left side of the big line. Will save you an hour.
Rome is a big tourist city, do don't fool yourself that you'll discover the one restaurant that tourists don't go to. But in the Rick Steves book he lists smaller places that are not tourist factories (remember, avoid anything that advertises "English Menu" in the window). We found a great spot for dinner in the Piazza Narvona IIRC - they have a open market during the day. Great gelato everywhere, try the dulce la leche. We enjoyed pizza at the many sidewalk cafes during the day and just watched the world walk buy. Note how easy it is to spot the American and UK tourists. Ugh.
For the Vatican which was pretty impressive, we splurged and hired a private guide. That was really helpful and we are so glad we did as he helped us navigate through crowds, pointed out all the cool stuff, history, etc. My kids really enjoyed it that way, otherwise it would have been a slog.
The Bath of Agrippa were a waste of time. I enjoyed the collesium a lot more than I thought I would, but the Forum was a little underwhelming (but you need to see it). I liked the Trevi fountain at day and night, but it was crowded. Pantheon was neat, as are most of the name brand sites. Here's an example of a Rick Steves
narrative and a note about
kids.
IMHO, looking at the Spanish Steps, you need to know the story about what you are looking at to see something other than steps covered with tourists. So that's where a great guidebook comes in handy (in advance) for helping you and your kid get the most out of the day.
Enjoy, and if I can answer any questions, please let me know.
Don