I've visited the area twice, both times in February. I stayed in Sorrento the first time, day-tripped to Pompeii, Herculaneum and Naples (the archaeology museum is a wow). A few years later I spent 5 days in Naples, then stayed in Salerno and rented a car for the Amalfi Coast and Paestum, though it's quite easy to use the bus for the AC and the bus or train to Paestum.
Transportation: To Sorrento, you take a fast train to Naples, then get the local metro (Circumvesuviana - CV) which takes 60-90 minutes. To Salerno you take a fast train all the way. The only ferries operating in the area winter are to Capri from Sorrento. If the sea is stormy, the ferry doesn't sail. There are buses to the AC villages run by SITA. You can download the schedules from their website - note that currently they are on the summer schedule; they'll be on the winter schedule when you're there. There are 2 lines along the coast, one from Sorrento to Amalfi, the other from Amalfi to Salerno. There is also a bus that does the climb from Amalfi to Ravello.
The weather is mild, can be very sunny, but you can also get rainy days, so you need to have a flexible schedule. If you're planning a short stay, you may be unlucky.
I would not recommend staying in Naples after only a week or so in Italy. It's not as tourist-friendly, to say the least. I enjoyed my stay there, but I was by then a seasoned traveler in Italy. Sorrento is a resort town. There's not much to see or do there, but it's a good base for day trips to Pompeii/Herculaneum/Naples on the CV, ferry to Capri, and bus to the AC. Restaurants are open, so are most shops, but there's no night life to speak of. The tourist shops, most lodgings, and many restaurants in AC villages will be closed, best to take picnic lunches since the few restaurants may be full, are relatively expensive, and kind of mediocre. The one I do highly recommend it Sal de Riso in Minori. You can get good lunches, pastries to die for, and excellent gelato. Salerno has lots of good restaurants, a well-preserved historic center, a large sandy beach and seaside promenade, and bars that are lively in the evening. It's also the cheapest for rooms and restaurants, outside of Naples. You can day trip to Pompeii by train - then a short taxi ride or a less-than-a-mile walk to the site and day trip to Naples by train.
Do not be deterred by the closures in the AC villages. The joy of the AC is the scenic beauty, not the towns.
