For seeing the AC itself and hiking, being on the AC is undoubtedly best. Note that some towns do not have ferry piers. There are 2 main bus lines - one from Sorrento to Amalfi, another from Amalfi to Salerno. Ravello is high on the cliffs above Amalfi, there is a separate bus line between them. Traffic on the road often moves very slowly - it's a narrow 2-lane road all along the AC, with lots of cars and tour buses. Someone mentioned the long lines for buses. Note that on intermediate stops, the number of passengers boarding will equal the number alighting. So waits can be long anywhere. You don't mention beaches. Most (all?) the beaches on the AC are very small and pebbly. My impression was that the towns mostly very small with few sights. The scenery is gorgeous.
I've visited the area twice, both times in February (no crowds, no ferries, no heat). I've stayed in Naples, Sorrento and Salerno. To visit Pompeii (and other ruins, like Herculaneum) or Capri, Sorrento and Naples are much better bases. You need the Circumvesuviana train (metro really) to Pompeii, Herculaneum. Naples is at one end of the line, Sorrento at the other. There are ferries to Capri from both cities. From the AC, you may have to change ferries in Sorrento. To the Circumvesuviana, you'll want the bus to Sorrento from the AC. BTW the National Archaeology Museum in Naples is a wow and there are other sights in the city as well. Salerno is at the eastern end of the AC. If you stay there, you can visit Pompeii by train and Paestum (ancient Greek temples) by train or bus. Another consideration is your budget. Both food and lodging are most expensive in the AC towns. Sorrento prices are lower, Salerno's even lower (and Salerno has a well-preserved historic center and a pretty big sandy beach and long seaside promenade), Naples probably the lowest.
If you are active and interested in sights, you may find very limited options for filling a week on the AC.
