I (not an insider) would not attempt both islands in one day. I think it’s better to choose just one at a time.
I am pretty sure that getting to either island involves either Sorrento or Naples.
There is a boat between Ischia and Capri, but it looks like it’s just once a day.
http://www.alilauro….
We only went to Capri. We hired some guy to take us around, and spent the entire day on the island, then ferried back to Sorrento.
I actually really enjoyed our visit, the funicular is cool. We skipped the Blue Grotto.

We stayed in Naples, and spent part of a day in Sorento, then stopped at Herculeum on our way back to Naples. Sorento was nice, but seemed very like parts of California. We spent a day on Capri, and it was very interesting, but crowded. We took a ferry before dawn to avoid the long lines at the funicular (said to often exceed an hour), so we saw a lot before the crowds showed up. We too skipped the Blue Grotto, as the water was choppy and we were late getting there (it is said to look best with calm waters and the sun at the right azimuth, but I’ve forgotten whether that means morning or afternoon (I’d guess morning)).

http://www.capri.net…

Has a lot of information on Capri and getting there. On the first page, there is a menu where you select your port to get the sailing times. I think there are four boats daily from Ischia. I would allow a whole day to see Capri both because there is a lot to see, and because there are a lot of people there to see it. There was even pushing and shoving just to get on a bus (we walked around a corner to the prior stop, which was not crowded). The side of the isle with the Krupp road and some gardens was particularly scenic, and not crowded, but then we were there pretty early.