I spent 4 days there pre covid- so hopefully someone with more recent info will come along. We caught the fast train from Naples, so not sure about travelling from the airport into Rome.
For the trains, I recall you had to validate the ticket for each journey. I think you buy the ticket- but need to use the machine to validate that particular journey.
A work colleague recommended trying the Jewish quarter for tasty, cheapish food. I recall a flattened, deep fried artichoke being delicious.
DH and I walked almost everywhere. Along the river and between all the main sights. We did get a local train/tube to the Vatican to see the Vatican museums. The amount of wealth on show was shocking, but its was equally amazing to see.
We visited in Late Oct/Nov, so didn't need to pre-book things, but I believe the queues can be dreadful in summer, so best to book ahead.
Word of note. DH and I have travelled extensively- but Rome was the only place where we felt we needed to be on alert a bit more. Mainly around the Spanish steps, but happens at all sorts of popular tourist locations. Don't engage if someone thrusts flowers/lavender/herbs at you- they will ask for money. Don't engage if they start wrapping a 'friendly' bracelet around your wrist- they will ask for money. Dont leave a mobile on your table if in a cafe etc. I know these scams can happen anywhere- its just the 1st time I've seen this 1st hand. I did, however, feel safe and wish we'd stayed longer there.