Myself and teenage daughter first time to Europe, 7 to 9 days. First idea was Italy, I would like water and some relaxation
Not sure if your daughter is closer to 13 or 19. It can make a difference in activities and interests. We took our 2 children several times in their teens to Europe and their favorite combination was: fly into Paris and see the sights for a couple of days, then fly (or train) to Nice to explore the Riviera. We love to stay in Antibes, a small town between Nice and Cannes, with beaches, a neat old town for exploring and good cafes, Picasso museum, close to Grasse and its perfume factory tours and the hilltop fortress villages of Eze and St. Paul de Vence. Italy is good, too. If I were going with a teen I would choose 2-3 days in Rome and then take the train to Sorrento for some seaside R&R, exploring Pompeii and the Amalfi coast as day trips. You can fly home out of Naples. Either trip would be nice for both of you and gives you a little time for cultural exploration as well as some vacation fun in the sun.
July. Daughter is 18
Also consider Capri. I originally didn't like the idea of Capri but now I love it.
What time of year are going?
Both Cinque Terra and Venice are perfect for relaxing by the ocean.
Monterosso has the only real beach in CT, but the five towns are close enough to visit the beach from any.
Venice isn't exactly a beach town but it's amazingly beautiful, on the water and surprisingly relaxing. Tourists throng there each day, evenings are great for strolling and casual dining.
I always travel shoulder season to save money and avoid crowds. Others on this site can tell you more about summer crowds.