I am staying for a week in Naples with my fifteen year old daughter for the last week of October and we'd also like to visit:

Pompeii (and Vesuvius, in one day?)
Capri and/or Ischia
Positano and/or Amalfi and/or coast but preferably all three! (have considered driver but seems beyond our budget, are there any reasonable tours or just use SITA bus?)
Rome

So any tips and recommendations for both Naples and these trips welcome, bearing in mind we will be using public transport (staying near Botanical garden) and are not mad keen on museums unless warranted.

Thanks in advance!

Your nearest metro stop is probably Cavour - you can take the metro to Garibaldi for connections to Pompeii and Ercolano Scavi (for a bus connection to Vesuvius). There is probably a bus route to Centrale/Garibaldi station - ask your hotel.

Metro Line 1 will get you to Municipio (the new stop) which is two minutes to the port for ferries to Ischia and Capri.

If you want to go to the Amalfi Coast, this is a long day on public transportation but I would opt for the train to Salerno and then take a bus from the 'opposite direction'. Ferries stop running in the last week of October but you could still find a service from Salerno to Amalfi - this is the best way to see the coast. You could return to Naples via Sorrento. Take the bus and then connect with the Circumvesuviana train all the way back to Naples to Garibaldi, then the metro to Cavour (or bus to your hotel whatever works).

Given an interest in Pompeii, the Museum of Archeology in Naples is something you might want to seriously consider visiting. I was glad I visited the museum after I visited Pompeii; other people prefer the opposite.

I agree with Blueeyedcod's recommendation for visiting the Amalfi Coast -- train to Salerno, bus along the coast, return via Circumvesuviana.

While on Capri, consider visiting the Villa San Michele -- lovely place, breathtaking views.

I found the Rough Guide particularly helpful for this area.

Herculaneum is a fantastic afternoon.

You might enjoy a late afternoon browse in Sorento with dinner there and the train back to Naples. You could do that at the end of the day coming back up the coast or in the evening after the visit to Pompeii or Herculaneum. Be sure to check late train times back to Naples.

Yes, agree with Sassafrass - dinner at Sorrento at the end of the day would be nice. Delfino on the waterfront is gorgeous. You could always take a ferry from Sorrento back to Naples - much nicer than the rattle-trap Circumvesuviana full of its assorted scoundrels and weirdos.