Caffé Sant’Eustacchio
Piazza di Sant’Eustacchio 82, 00186, Roma
The best coffee you can have is at Caffé Sant’Eustacchio, often full of people but they are quick in serving, it is near Piazza Navona and Pantheon which are both must-sees for newcomers to the city, so you’ll easily find yourself in the area.

via degli Uffici del Vicario 40, 00186, Roma
The best ice-cream can be found, without a doubt, at Giolitti. The cue here often scares people away, but don’t be, sit at a table rather than taking the ice cream to go and it will be a nice stop to rest after having walked throughout the day. Giolitti is in the city center, not far from Caffé Sant’Eustacchio, so you will find yourself in the area quite easily.

Babington’s Tea House
Piazza di Spagna 23, 00187, Roma
Right to the side of the famous steps of Piazza di Spagna. Please beware the prices here are hefty but I recommend you stop by even if you are not planning to eat or drink anything as it is a very interesting place to see, so do have a walk inside.

Romolo alla Mole Adriana
Vicolo del Campanile 20, 00139, Roma
If you are visiting the Vatican and you are looking for typical Roman dishes, this is a great place where to stop, not far from Basilica di San Pietro.

Osteria delle Commari
Via Santamaura, 33-5, 00192 Roma RM
This place is 300m from the Vatican Museum, it is a great place where to have a bite to eat either before or after your visit. Again, this is a very typical place where to have traditional Roman dishes.

Hostaria da Benito e Gilberto al Falco
Borgo Vittorio, 34a, 00193 Roma RM
Close to Basilica di San Pietro in the Vatican area, this place is renowned for their fish dishes.

Osteria Barberini
Via della Purificazione, 53-15, 00187 Roma RM
Osteria Barberini is 10min away from Piazza di Spagna (walking), so right in the city center. It is a great place to stop and eat if you are taking the tour of the center and stopping at Piazza di Spangna.

Amedeo Ristorante
Via Principe Amedeo, 16, 00185 Roma RM
This restaurant is close to the Roma Termini train station, which makes it a great place where to go and eat before catching the train to another destination. The food, again, is typical from Rome and the service is fast.

Trattoria Monti
Via di S. Vito, 13, 00185 Roma RM
Similar to the previous one, this restaurant is close to the Roma Termini train station, the food is amazing and another great option to have a bite before catching a train.

What can't be missed:

Chiesa San Luigi dei Francesi
Piazza di S. Luigi de' Francesi, 00186 Roma RM
In the same area of the Pantheon you can find Chiesa di San Luigi dei Francesi, in it you can see a masterpiece by Caravaggio.
As they celebrate mass in this church it can be visited in the following hours only:
Mon-Fri: 9.30 - 12:45 or 14.30 - 18.30,
Saturday: 9:30 - 12:15 or 14.30 - 18.30,
Sunday: 11:30 - 12:45 or 14:30 - 18:30.
It is worth a visit as not only you get to see Caravaggio’s work where it’s meant for it to be.

Santa Maria della Concezione dei Cappuccini
Via Vittorio Veneto, 27, 00187 Roma RM
Here you can also see more of Caravaggio's work, moreover you can also visit the Cripta dei Cappuccini, which can be quite interesting. Before visiting the Cripta I suggest you run a Google search in case it could disturb you.