Rome will be very hot in July. So book The Forum/Palletine Hill and the forum early as you can. Even if you can’t get a tour below the surface of the collisseum you’ll still love it (I’ve been a few times and not got in).
Spend your first day just wandering. It’s very walkable, just be careful crossing roads! Follow a nun, they say. Go to the Trevi fountain early, wander through the streets to the Spanish steps, Piazza Narvona, the Pantheon, Castel st Angelo, St Peter’s square, Trastevere, Campi dei fiori and the Vittoro Emanuelle building (which if you climb up gives you a great view down into the forum. Lovely for sunset.
Book your Roman day for day two. Take snacks and water with you..
Day three. Decide what you prefer. I love the Roman side of Rome, but the Vatican side doesn’t do it for me as much. I was hugely underwhelmed by the Vatican museums and the Sistine chapels, and I’d never visit again (whereas I definitely would the forum), but I did like inside the basilica of St Peter’s, particularly the climb up to the roof, which gave a good view. If you’re not that bothered, then perhaps Villa Borghese and a wander around the park may be a good option.
I wouldn’t be booking trips out to Pompeii etc on a three day stay - come back for that, it is worth it.
Enjoy it, it’s a wonderful place. Don’t be over worried about pickpockets, it’s not worse than anywhere else. Be sensible, be aware of what’s around you. Beware of people trying “to help you” at stations or tourist spots. They can be scammers or distractions for pickpockets.