We went to Rome in June, and it was indeed very hot, especially when going round the Forum with no shade. I took a telescopic brolly to use as a sunshade.
July and August will be hotter - go in June if you can.
I bought some guide books,like the AA Essential Guide and planned what we would see each day. ( we went Monday to Friday ).I pre booked morning tickets for the Vatican Museum online, as there are very long queues for this. ( tip from Trip Advisor Rome forum ) After the museum we went to the top of St Peter's Basilica - lots of steps, but well worth the climb.
Trevi Fountain, ( beautiful ) Spanish Steps, ( they only have flowers on the Steps in May ) Castel Sant'Angelo, ( more climbing ), Pantheon ( amazing building ), Elephant with obelisk outside Santa Maria Sopraminerva church, Piazza Navona and other Piazzas, Colosseum and the Roman Forum ( buy a dual ticket from the Forum to save a queue at the Colosseum - another tip from the TA forum ), Sistine Capel and Vatican City Museums, St Peter's Basilica, The Mouth of Truth, and so much more.
It is lovely just wandering around the city at night, and taking everything in, or enjoying a meal on a Piazza, where musicians play for money.
I definitely want to go again one day.