I can recommend staying at the boutique Hotel Alpi - I loved the interieur and the breakfast there. And the Trastevere District as well as the Jewish Quartier are definitely worthwhile your time.
Take an early morning hike on the Spanish Steps and enjoy the view of the Fontana del Barcaccia on the Piazza di Spagna and the famous Via Condotti before its covered by crowds. Rent a scooter and come back in the evening for a completely different impression.
Enjoy a picknick in the park Villa Borghese and ride tandem bikes rented at the Via Veneto entrance.
Fontana di Trevi by night
Capitoline Hill, Cordonata Steps and Santa Maria in Aracoeli
With a husband that loves to cook and fill his kitchen with all sorts of gadgets - of course you visit c.u.c.i.n.a when you are in Rome and have espresso at Gusto next door.
Via del Corso is long and features main stream shopping the italian way! Half way down you find La Rhinascente - the best known department store with both local and international brands.
Experience the sunlight floating into the Pantheon - my favourite building in Rome. And cool down with delicious hand-made Italian Gelato at the Giolitti afterwards.
Area Archeologica del Teatro di Marcello
The Vatican Museums is busy but very interesting in particular with a guide.
Rent a scooter and don't miss driving on the curvy green Via Vitorio Veneto - remember the cool italian helmets!
Before dinner enjoy the sunset scenery around Colosseo and Cherub Fountain.
Rent comfy bikes at Roma Rent Bike who also has really good guides that will show you Rome from interesting angles.
Piazza Navona and Fontana del Moro, Fontana del Nettun and Chiesa Sant' Agnes in Agone.
The scenic view from Piazza Napoleone I in Villa Borghese. Below you find Piazza del Popolo a famous location in the movie "Roman Holiday".
If you fancy a challanging bike ride go visit The Knights of Malta's Keyhole and don't skip biking around Circus Maximus or crossing the English Bridge to the Bocca della Verita (Mouth of Truth).