I think three weeks is a good length - I wouldn't go any shorter - and it should give you time. I would think something like:
Rome 5/6 nights which is 4/5 days. There is so much to see planning Rome is hard. (I'm planning 5 nights right now and there isn't enough time.)
Naples 3/4 nights - see the city, Herculaneum if you want and maybe Pompeii
Salerno 4/5 nights - see Paestum, Pompeii (if you haven't) AC
If you want to stay on the Amalfi coast you will need arrange that time. So that's about 2 weeks on its own and then add Calabria and probably a full day to get back to Rome or Naples to fly home. If Calabria is only a few days you could expand your time above.
Possible additions: Ostia Antica (Roman ghost town easy to reach from Rome), Caserta - an important historical town near Naples with no American tourists.
Calabria is 3-4 hours on the train from Salerno depending on where you are going. If you need a car you'll need to plan where to pick it up and return it since much of Calabria are small towns.
Others will have other suggestions or timings these are just my opinions.
My tips:
Salerno has a wide pedestrian shopping street Corso Vittorio Emanuele that runs from the train station to downtown. I reccomend staying anywhere in this area for easy access to strolling and the train station. Once you hit the old town it breaks down into a series of old streets, alleys and passageways. Restaurants and bars are scattered all over in this area. The cathedral - especially the basement - is worth touring and search the oldest parts of old town for murals and art alleys.
If you're in Salerno for an extended period of time do not miss the eté Prime at 174 Corso Vittorio Emanuele tucked away between the clothes shops and bank. The front opening onto Corso Vittorio Emanuele (the wide pedestrian only shopping street) is small hallway but set back from the street is an amazing grocery store worth knowing about. The store is kind of like Conad meets Whole Foods and one in Salerno has high end goods at really good prices. I saw 6 kinds of local specialty Buffalo Mozzarella for 2-6e, selection of bakery breads, and shelves of Barolo starting at 12e and topping out at 23e.
The many fruit and daily needs places that dot the street are fine for everyday things but if you have a kitchen or want to put on a real spread and a nice bottle of wine this store is a great place.
A minor tip but: The best sfogliatelle in the world is 5 minutes from the Naples train station. Antico Forno delle Sfogliatelle Calde Fratelli Attanasio Vico at Ferrovia 1/2/3/4. Walk over, show up, grab a number right away and wait your turn. So worth it.
Enjoy sud di Italia,
=Tod