I'm getting repetive here but I'm a great fan of largo de Torre Argentina. Best gelato place , pizzeria Florida pizza to take out by the slice and a cat sanctuary with 4 temple ruins. Fabulous and the kids will love it.
About 2 blocks away there is the crypto balbi which is a tiny museum packed with a pretty effective introduction to the timeline of Rome that would suit the children well, I'd be tempted to do that first . Apparently if you get the day right ( I didn't ) there is an outdoor guided visit to and extremely well preserved latrine complex!
Palazzo altemps is another small but excellent museum it's on a combined ticket with the crypto balbi and palazzo massimo ( also excellent with complete " room sets" on the top floor - near termini).
The 3 small museums don't need pre booking but a skip the queue for the coliseum and the forum as early as you can face getting there. The huge benefit of being in the centre is to do the big attractions before they get busy - they can be booked on line.
Realistically you'll spend loads of time in gardens ! The villa borghese gardens of course are lovely but if you want to go in the villa borghese read up carefully in how it works and pre book. They let the party in on the hour then shut the doors and chase you through lol
Pantheon - of course it is amazing. Spot the elephant and obelisk statue in the next square too, as well as granita at the golden cup coffee house ( s it Tassa d'oro or something like that )
A day out at Ostia antica ( local train from pyramide) would be fun huge site explore but might be very hot.
Do not do what I did and walk from pyramide to the baths of Caracalla. They were immense and amazing but I nearly died in the heat at Easter !