Italy w Teens late June

RE round-trip Rome: I'd look at southern Tuscany, which sounds perfect for your group, and you could potentially keep it to one base (depending on arrival and departure times), which would be great with just a week.
There are a lot of outdoor activities in this area (Giglio island, rock and sand natural beaches, the Orbetello lagoon, Saturnia hot springs, Maremma park with ebikes and huge cows, "vie cave") plus some educational but cool stuff like Etruscan archaeological sites to entertain you. I stay in towns usually so I cannot recommend a specific agriturismo stay, but this area could offer everything you are seeking.

Fly to Rome (FCO), rent a car upon arrival. Picking up the car in the city might save some money, but when comparing prices consider the 55€ for the taxi ride from the airport to the city, actually a bit more given 5 people riding.

Use www.autoeurope.com for rental car pricing. Enter pick up ROME ALL LOCATIONS and return at the same pick up location.

Hopefully you won't be too jet lagged, but drive carefully.

Head to South-East Tuscany area (Montepulciano, Pienza, Val D'Orcia), which is one of the most classic of the Tuscan locations and close enough to Rome (less than 3 hours, mostly freeway) or also the Coastal area of southern Tuscany called Maremma.

After spending let's say 4 nights at an agriturismo in that area, drive back to Rome. You must spend at least one night before your flight in Rome to make the morning flight. If you have never visited Rome, then a bare minimum or 3 nights is recommended. In that case, it is best to stay in a hotel in the city center. You can return the car at a downtown office. Roma Termini station or Roma Tiburtina station are relatively easy to drive to, without risking to end into the ZTL (Limited Traffic Zone) of Rome, where permits are required to drive.

If you have never driven there learn the International Road Signage in use in Europe.
https://en.wikipedia.org/wiki/Road_signs_in_Italy
Become familiar with limited traffic zone concept:
https://www.autoeurope.eu/travel-blog/what-are-ztl-zones-in-italy/
be aware that traffic laws are mostly enforced by hidden cameras (well over 10,000 cameras immortalize traffic violators 24/7)

We took teenage grandkids to Italy for two weeks and it was a very successful trip!
Choose a central location in Tuscany to stay. We like staying in the towns between Siena and Florence. Look at Panzano! We have rented from Chianti and More, nice properties, and very good service.

Just a caveat that many agriturismos have a firm Saturday-to-Saturday stay requirement, so that will narrow your options somewhat.

If the teens dislike shopping, crowds and multiple museums, Rome and Florence might not be the best fit. Cinque Terre in the summer is the very definition of masses of people moving together along the trails, so I'd skip that too. You could pair time in the Tuscan countryside with a few days at Forte dei Marmi, which is known for its beach clubs, or Elba, which is more local. Or fly to Sardinia for its beautiful beaches.

We lived in Germany for four years and had teens we took with us when we toured Italy and other countries.
They were as interested in the history and culture as we were. We didn't try to kid down our tour.

We did try the beach at Rimini on the NE coast, but the beach was not very nice at all.

Tuscany is loaded with wonderful smaller cities and towns like Siena, Lucca, Pisa, San Giminigaco(sp).

Also, many posters on this forum that rented cars in Italy have gotten expensive tickets trying to drive into cities and towns that have restrictive entrance policies.