I am just back from ten days there...
Firstly, the pickpocket advice is solid - but also make sure if you are sitting down to eat outside anywhere that you have the straps of your bags wrapped around your feet / legs - I saw several bags get stolen this way.
Take a water bottle, there are water fountains everywhere and water in the tourist areas is expensive, plus cafes won't fill water bottles up with tap water.
Eating down the back streets tends to lead to better food and
I would not recommend booking a tour at the Vatican Museums - take the audioguides instead. The tours rush you through as they are a money making experience and try to cram in as many tours in one day. Do however book priority line access (that is worth the money!).
St Peter's is free but the best time to arrive is 07.00 in the morning when there is no line - head straight up to the dome (which you have to pay for, either lift or none lift, having done the latter I would recommend the lift - don't worry there's still 500 stairs to go). Take water. Later in the day the viewing platforms becomes so crammed you can't actually experience the views.
The light show at the Forum was a lovely experience, and we booked Colleseum tickets that actually mean you can do the Forum etc (not the light show, you book that separately) for the following 24 hours. Again, we booked priority for this but no guide and used audio guides.
Taxi from the airport seems pointless - take the express train or the regular train - and if you want buy a metro card for the week (around 25 Eu) but can be used across all networks. I found it quite hot so would walk all morning, then use PT back to hotel for a rest / cool down (and most things close) and then use a combination of PT and walking in the evenings. PT was excellent for late night returns.
Definitely go to Villa Borghese, Botanical Gardens, and Villa Doria Pamphili and Villa Medici if you like gardens.
Churches are free but I was quite shocked by the number of people who didn't cover shoulders or legs - and who did things like filming people praying.
Basilica Papale di Santa Maria Maggiore, and Basilica di San Giovanni in Laterano were my favourites by far.
You can get direct from the Sistine Chapel into St Peters, avoiding the queue for St Peters. This option was not open when we were there and has apparently not been open since covid.
Check when cruise ships are coming in as it makes the main attractions much busier - Wednesday and Thursday were the bad days a couple of weeks ago.