Galleria Borghese is fantastic but you must book in advance.
If you are not doing the Vatican Museum tour (pre booked) then be prepared to queue for approx an hour to access St Peter's directly, as you have to pass thru airport style security.
After the Colosseum, make sure you go up onto Palatine Hill and into the Forum (should all be included in one ticket).
Recommend San Ignazio church near the Pantheon for the amazing ceiling.
Have coffee at Cafe Greco on Via Condotti (closed Wednesdays).
Go down to the market at Campo Fiori and walk around the nearby streets then up to the Theatre of Marcellus.
Go over the river to Trastevere and just wander around the little streets, stopping for coffee (but never order cappuccino after late morning, as it marks you out as a tourist 🤣).
You don't need to use public transport at all, but I second that comfy shoes are essential.