In August you won't find many Romans - it's my fav time in Rome. September will have good weather. April and May will be very good. But the city - my city! - is fantastic all year long. Full of tourists all year long, too - so perhaps it's better to avoid Christmas, Easter etc.
Go to Trastevere, Ponte Sisto, Piazza Farnese, Campo de' Fiori. Have breakfast at the San Calisto Bar near Santa Maria. Go to via dei SS Quattro near San Giovanni. Visit San Clement and its Crypt. Get lost in the city center: it is huge and fascinating. Perhaps go visit Villa Adriana.
Have dinner somewhere nice in one of these places: