Went to Almafi coast back in November with my wife and thoroughly enjoyed it. Its full of culture and its very charming. I also recommend using Sorrento as your main hub as transportation from Sorrento to Positano, Almafi Coast, etc. From Rome to Sorrento is a 1hr train ride if I remember correctly and its the last stop. We stayed at Palazzo Jannuzzi Relais for only 4 nts, which was at the central square where you had the 100 stairs leading to the dock for ferries. Due to our short stay, we had a packed itinerary. I think what we did was do a bus ride from Sorrento>Positano>Almafi Coast starting early morning stoping at each town to casually walk it. The bus stop was right in front of the train station, arrive early so you get a good seat. Roads are windy and I've heard of ppl getting motion sickness. Each town is different in its own way so make sure you give yourself enough time to walk it and snap pics. On the way back, we jumped on the ferry from Almafi Coast back to Sorrento. The nice part of that was the breathtaking view of the towns from the sea. Since you're staying longer, you could do day trips to each town and spend time on the beaches. Positano was my favorite of the 3. In Sorrento, we ate at Inn Buffalito and some seafood restaurant in the Marina area (Take a cab or bus if you go).
When you go to Capri, make sure you do the blue grotto. Tip the man that rows you into the grotto so you get to spend a bit more time in there. Well worth it. From there, ask to be dropped off at the steps next to the grotto, where you can jump on a bus to the tram that takes you to the top. This saves you a lot of time as you don't have to head back into the marina to jump on a bus. Think of it as going counter clockwise on the island.
In Naples, make sure to stop at Pizzeria Da Michele. Pizza in Naples is a must and we made a pit stop before heading to Rome. Its far easier to jump in a cab with all your luggage than to wheel it there. Its a 15 minute walk from the train station. Well worth the hassle for a Margherita.
Some of my research was done on trip advisor forums but most of it was from a Rick Steves book. Definitely pick one up as it highlights the must do's and also provides dining recommendations.