For the best views, start the Path of the Gods from Bomerano, above the town of Amalfi, and walk towards Nocelle. It’s accessible via bus from Positano to Amalfi.
In Capri there are great Airbnb choices in Anacapri, further up the mountain from Capri’s centre. It’s less crowded, loaded with trendy shops and has plenty of nearby bars.
For something different, head outside of Rome for an authentic cooking class. Convivio Rome is a cooking school an hour train ride from Roma Trastevere station to Fara Sabina. You’ll be picked up there. A full day class is €129 (about $A195) per person.