12 Day Trip (Max) to Paris, Florence, Rome, and Positano

Check every iteration--into Paris and out of Naples, etc. These are usually called "multi-city" searches. I use Kayak and Google for searching, but I book with the airline. Book flights when you get a good fare you can live with--with a better schedule.
For accommodations, I use Booking.com, where you can search by budget and all other parameters. Its map function is useful.
With that time frame, you'll have to fly from Paris to a city in Italy. While there are low-cost carriers, they often have strict baggage limits. You might consider saving Paris for a future trip to cut down on transport costs and save time. Within Italy, train travel is a snap. Search on Trenitalia and Italotren websites. If your schedule is set, you can save a bit of money by booking in advance, but train tickets generally won't break the bank.
For Positano, you can take a train to Naples or Salerno (it depends on where you come from), and ferry from Salerno is the best way to arrive at Positano.
Book asap, especially for Positano--it's an expensive popular place. All of the Amalfi coast villages are pretty, and they are linked by ferry. Do search others as accommodations may be less expensive elsewhere.

Ah to be young and adventure driven! You're really asking a lot here and you will receive lots of excellent advice/ideas. We need to get some clarity: are you counting your departure day and return day as part of the 12? That would bring you down to 10 but remember you will have an overnight flight so your first day on the continent will be shorter than you might think.
Plane is probably the easiest from Paris to Florence. After Florence, train to Rome. OR Do Italy first and fly home via Paris? Check schedules and fares to help you decide. That said, train and bus is the best way to move around Italy.
Positano? Really? First, it is a quite long way from Rome to Positano. The Amalfi Coast is indeed beautiful, and extremely crowded in June, but you'd be looking at a whole lot of time on trains and buses - in both directions.
I've always stayed in VRBO apartments in Paris, Florence and Rome. You can save a lot of money that way since hotel prices will be at peak.
You can book your museum tickets much closer to June but now is not too early to watch airfares. Google/flights is a good resource. If you can be flexible with your dates, it might help.

Spend a lot of time with the planning. Make lists of all the things you want to see, do and experience. That will help you figure out how much time in each city. To be honest: for the 12 days (10? 9 1/2?) that you have, I'd do Florence and Rome only. But that's me.
Julianna, welcome and enjoy the planning! It will help you enjoy the trip all the more.

Julianna, welcome to the Forum. You have received excellent advice so far about logistics. I too, wonder how many actual NIGHTS you have for this trip. It will be a whirlwind and I’m sure you and your friend will have a great celebration trip.

Have either of you stayed in hostels? They are a great way to connect with other worldly travelers and can be much more affordable than hotel or apartment stays. Many hostels now have private rooms available (along with dorm rooms with shared bath). I have stayed at The Beehive in Rome twice, many years ago (2010 and 2012) and loved my time there. It is near Termini train station and although some posters on this Forum will caution travelers about staying near train stations, I was not insecure about the location and I was a solo traveler both times I stayed there. Linda and Steve are wonderful hosts and genuinely nice people. Here is a link:

https://www.the-beehive.com/