Florence and Rome for a week

Any tips, recommendations are appreciated!

I would plan 2 'vacation days' at home when you return to recover.

I would do the Pompeii/Mt Vesuvius on the 8th. After doing that the next day could be a a real struggle to get moving, so just sitting on the plane may be a welcome relief.

You have 6 days, 3 of which involve some traveling. Do you have some ideas about what you want to do in the places you’ve listed? It’s way too much for the time you have, but I assume the flights have been purchased. So-o, I suggest you cut out Pisa and CT. Have either another day in Florence or another in Rome.
You’ve planned a day trip from Rome, so stick with that, if Pompeii is important to your group. Personally, I would prefer Orvieto.
You can maximize your time in Rome by doing a couple of guided walking tours. There are many recommended on this site. You would be amazed at how much time can be eaten up trying to find your way around a megalopolis like Rome, but a guide knows the most efficient ways to structure a tour. The cost will pay big dividends in everyone’s enjoyment
You might want to look into tours that are geared to young folks.

Please forgive my recommendation to reconsider your entire plan.
Having been to all of these places, and with a very clear understanding of what you can experience I would suggest a more pleasant alternative.
Fly to Rome. Forget Pisa, Cinque Terre, Florence and Pompeii.
Replace your day trip to Cinque Terre with a trip to Orvieto (by train).
Visit Ostia Antica instead of Pompeii
Spend the rest of your time in Rome where a lifetime is barely enough to take it all in.

If you have your heart set on Florence, do that as a separate trip and plan two or three days there. Do day trips from there to Sienna, wherever. Then travel to Pisa (if you must) on your way to Cinque Terre where you should spend several days.
If you must visit Pompeii and Vesuvius do it on a Southern trip where you travel from Naples to the surrounding areas while on your way farther South.
Ostia should be a more interesting experience than Pompeii but we all look at things in our own way.

You will spend an enormous amount of time on a train doing nothing but watching the country side. For us, that is a killer on a short time trip. You could do Pisa as day trip from Florence, CT is too far for a day trip especially if you divert to Pisa. Skip it. Pompell and Mt Vesuvius is also out. Pompell is barely doable as a day trip from Rome. Adding Vesuvius is out of the question. Save it for another trip. And Mack has a good recommendation.

I have to agree with Mack’s recommendations.
With limited time I also suggest Rome and a possible day trip to Ostia Antica. I also like the idea to take a day trip to Orvieto. Less than 2 hrs from Rome. Orvieto is one of the most enjoyable Hill towns in central Italy. Overall “less is more.” Enjoy your planning.

Another vote here for Mack's plan. Pisa + the CT is not a reasonable day trip from Florence. Pompeii + Vesuvius is not a reasonable day trip from Rome.

Mack's plan:
Fly to Rome. Forget Pisa, Cinque Terre, Florence and Pompeii.
Replace your day trip to Cinque Terre with a trip to Orvieto (by train).
Visit Ostia Antica instead of Pompeii
Spend the rest of your time in Rome where a lifetime is barely enough to take it all in.

I think you really only have time for 2. 3 nights Florence and 4 nights Rome. Leave the rest fir another trip, the first time we went we did 3 cities, and had to choose. We didn’t do Florence, and are doing it in May. You need to just enjoy yourself instead of running ragged,

OK, you asked.

I agree that there is way too much scheduled here to really see anything at all.
I would stay in Florence for three nights; and if you really want a day trip from there, Pisa AND CT are way too much for one day.
I would suggest a day trip to Lucca instead, it's only 1hr 40 mins on the train, and is just a delight...biking round the walls, walking around inside the town.
Your teens will like it as it's different from anywhere else.
Then get to Rome for your last four nights.
Forget Pompeii this trip.
If you want to see ruins that are similar, Ostia Antica is very near Rome and is very similar.
There is so much to see in Rome itself , you will barely scratch the surface.

If you spend most of your days on trains, you won't see anything at all.
Italy is meant to be enjoyed, not checked off a list.
You can always return!!
And you will!

I like fast travel, but this would be too fast even for me.

I agree with the suggestion to leave out Cinque Terre. My suggestion would be:

4/2 - arrive in Florence late afternoon (approx 2:30 pm @airport) - hotel Degli Orafi. Walk around. Get gelato. Figure out the lay of the land.
4/3 - Florence. Depends what you want to see here. Options include the Uffizi Gallery (Boticelli's Birth of Venus, et al), the Accademia Gallery (Statue of David), the Duomo/tower etc., the Pitti Palace and Boboli Gardens (I enjoyed these but would probably skip this if I had such a limited time.) Of course, there's the Ponte Vecchio, but you pretty much can't miss it, and the Piazza di Signoria, which is easy to see as it's in the same general area as the Uffizi gallery and the Duomo.
4/4 - Pisa day trip, plus some time wandering Florence in the evening.
4/5 - high speed train to Rome (time TBD) . Rome sightseeing after you get there. You could walk around and see the Pantheon and Piazza Navona, or the Trevi Fountain and the Spanish Steps, for instance.
4/6 - Rome sightseeing. Colosseum and Palatine Hill, perhaps. Also wander and eat. :)
4/7 -Pompeii / Mt Vesuvius (day trip) --look for a tour to Pompeii and Sorrento with the Roman Guy. We did it, and it was great, albeit to Pompeii and Sorrento. (Note that you see Mt Vesuvius on the way and from Pompeii. If you want to climb it, then you would probably want to do an overnight in the area.) Alternatively, you can take the train to Naples and get to Pompeii from there. Some people say this is faster, but for us, we liked the simplicity of the tour. We also liked getting to spend a few hours in Sorrento--a beautiful town that might help to make up for not seeing Cinque Terra. It IS a long day and a long drive, but if you're prepared for that, it's still worth it, IMO.
4/8 - Rome sightseeing. Vatican and St. Peter's perhaps.

Tours we did in Rome included the Capuchin Crypts (loved), the Appian Way, and the Catacombs (wouldn't do again) with the Roman Guy, and also the Vatican Museums, Sistine Chapel, and St. Peter's Basilica with the Roman Guy.
4/9 - Travel home

Appreciate all the responses. We have already purchased air tickets so we will be going Rome and Florence however I will reconsider the day trips and look into your other suggestions. Thanks much!

https://community.ricksteves.com/travel-forum/italy/pompeii-and-vesuvius-69e8366c-1e75-4a14-a0bc-0677269492d1

https://community.ricksteves.com/travel-forum/italy/pompeii-day-trip-from-rome-2c38f926-9036-480d-95fd-403dd7694e4a

I know this has been mentioned before, but I wanted to second/third Ostia Antica. I understand that the museum at Pompeii has recently reopened, and it's supposed to be wonderful. But, if you're just up for the Roman ruins experience rather than specifically wanting to go only to Pompeii, Ostia would be a better choice. A generous 3 hours to Ostia, including the transportation time, makes for a nice experience and would allow you to spend more time in Rome. Ostia has a charming small museum, and the cafeteria is ok. Not great, but a good way to grab a bite before moving on to the next event.

Mack got it right with his comments.
Minimize your travel more.

The Cinque Terre is no day trip from Florence so skip it.

Way too much! Slow down. That will mean cutting things -- frankly, I'd cut it down to Rome and Florence. You can busy yourself for a week in those two cities and still want much more. Zipping to Pompeii and Pisa/Cinque Terre (both in one day?) will mean you do nothing but transit from one place to the next.