Rome: so, you don’t have much time. Don’t waste it wandering aimlessly through Trastevere’s tourist traps or throngs of clubbing kids on Friday night (there’s good reason to visit this beautiful neighborhood, but there’s also that). Also, don’t waste any meal hoping you’ll just happen to find a worthwhile place. You might. But you’ll need a little luck.
Take a nap when you arrive if you need to, but get going as soon as you’re able. You’ll wake up, trust me!
Friday: For your first night in Rome, I’d recommend casual cucina Romana, then a long, late-night, after-dinner walk through the centro storico. (especially in August, daytime is the worst time to visit the major monuments, so do it after midnight!). Make your first night special as you don’t have nearly enough time to do anything but scratch the surface of the eternal city.
You might start the evening early with a negroni or aperol spritz at the gorgeous bar in the Hotel Hassler. You’re right at the Spanish Steps, so head on down to the Piazza di Spagna afterwards and head directly into the heart of the Centro Storico. The nightly passeggiata on Via del Corso will be in full swing.
Consider Armando al Pantheon http://www.armandoalpantheon.it/ for dinner (get reservations). Absolutely classic, utterly unpretentious, as popular with the locals as the tourists. In addition to the classic roman pastas (carbonara, l’amatriciana, gricia, cacio e pepe), you can try excellent versions of quinto quarto classics like trippa alla romana or rigatoni con la pajata.
Or you might start on the other side of central Rome in Trastevere with wine at a nice enoteca or negroni or aperol spritz on Piazza di Santa Maria. Then jump across the river to the Campo and consider eating at Salumeria Roscioli http://www.salumeriaroscioli.com/restaurant/ for classic Roman pasta dishes, hams and salumi, wine, AND NOTHING ELSE. Get reservations. After a long lingering dinner, stroll through the Campo di Fiore, then keep going to the Piazza Navona, Pantheon, Trevi Fountain.
Get lost. You’re never really far from where you might want to be. Just wander and be as mindful of the nooks and crannies as you are of the glorious Bernini fountains.
Sunday: it’s all about the lunch, so much so that many restaurants aren’t open for dinner. Again, cucina Romana! Since you’re visiting the Coliseum that morning, I’d definitely consider lunch in Testaccio which is just south of the ruins (one Metro stop if you’re using public transportation). I’d recommend something like Flavio al Velavevodetto http://www.flavioalvelavevodetto.it/ or Perilli (last meal before the rapture/earthquake/end of days: perilli - Elizabeth Minchilli).
On Sunday night, pizza is often the late-night default option. Maybe Emma Pizzeria Con Cucina to sample the other classic Roman pizza style - cracker thin crust - since you’re probably going to Pizzarium for pizza al taglio.
And, again, late at night is when you want to stroll the city, in the relative cool with fewer crowds. The monuments are lit up gloriously. This is when the magic of this city permeates the air, when rounding a dark corner can bring you face to face with some astonishingly grand beauty … if you go late enough, you can have it almost all to yourself.