If this is going to be your first time in Italy @SellingStevenage then I agree to keep it simple and just do Rome and maybe a day trip or two to places nearby. Ostia Antica was the summer resort of Romans in Roman times and rather go schlepping off Pompeii, this is where I would recommend you go to get a taste of Ancient Rome. If you love Italy and decide to return, then Naples, the Amalfi Coast, Pompeii, Herculaneum, etc will all be there for another trip, but I wouldn't try to do any of that this time.
I loved Trastevere when I first visited Rome, but we stayed there in an Airbnb with our kids a couple of years ago and I was shocked at how grotty it has become - lots of empty stores, lots of graffiti, lots of gangs of roaming youths - I wouldn't rush back. The Jewish Quarter was lovely for a wander, but you'll find lots of nice hotels in central areas and if you're going with a tour operator I'm sure they'll be able to recommend a suitable one. Don't stay slap bang next to a big tourist attraction like the Trevi Fountain as it's noisy and crowded day and night.
The Vatican is a must. Book tickets in advance for the first entry slot of the day and when you get in, speed walk to the Sistine Chapel as your first port of call and you'll get it entirely to yourselves! St Peters and the Vatican Museums are amazing and will take an entire day, so don't rush, buy a guide book, take your time and enjoy.
But there is so much to see and enjoy in Rome. Just off the top of my head: the Vatican, Spanish Steps, Trevi Fountain, Pantheon, Forum, Colosseum, Arch of Constantine, Trajan's column, Piazza Navona, Capitoline museums, Museo Doria Pamphili, the catacombs and the Appian Way, Villa D'Este, Villa Borghese, Ostia Antica, and many, many more Roman monuments, shops, ice creams ... see you have more than you'll ever see in one week. I've been to Rome about four times and I still haven't seen everything!