Went last August, last minute. Had no problems booking anything other than the underground Colosseum tour - but saw the whole of the rest of the Colosseo. The Colosseum and Forum are in the same ticket so I’d recommend doing the C first then having a decent bite to eat and tackling the Forum, once you’re in you’re in, there’s no return entry once you leave and limited food options inside. The Forum is fantastic though, and huge. It gets boiling in the middle of the day. one option is to do the forum first thing and the Colosseum in the early evening.
the other must see is the Capitoline museum, I don’t think you get everything out of the forum without the context available in the Capitoline.
I would forego the Vatican altogether unless you have a specific connection to the Catholic faith. It’s expensive, and crowded. The Raphaels are good. The rest is duplication. The Sistine chapel is an unpleasant experience - crowded with stressy guards. It’s not worth it for a 3 day trip.
The Castle is very good. We went in on a whim and that was a highlight.
some of the Basilicas are incredible. I really like the Basilica Santa Maria Maggiore, and didn’t manage the Basilica San Clemente but regret that.
Nero’s palace is open on some days only and worth a try.
the Borghese gardens are worth a visit with kids, we did the very touristy little row boat thing with the turtles in the pond.
there is so much to do, just soak up the atmosphere. I wore a full on bumbag and had no problem with thieves. You will need passports to enter some places, so that’s a must.