Ooh ooh I love Ischia. Yes you must go to the Mortella botanical garden. It's very quirky, and has a nice tea room. It's bloody enormous though, it took ages to get round it all, but it's worth it for the views from the top over the amphitheatre.
We also hiked up Mount Epomeo, which was really nice, there is a restaurant at the top. You also see all of the Ischian bunny rabbits running about before they turn into the islands signature dish- Ischian Rabbit, which you get in loads of restaurants.
You must go to castello aragonese, which is on it's own little tiny island with a walkway up to it. It also has a weird medieval sexual torture chamber in it with loads of olden days BDSM type stuff in it which was...interesting.
We loved Ischia. Our hotel had a natural hot spring pool in it which was really cool and went spent lots of time bobbing about in the orange water. We also hardly saw any English tourists which was great, it felt far more authentically Italian that places like Lake Garda that always have loads of English and German tourists.
We did find that not that many people spoke English, so get an app dowloaded in case you need to translate from English to Italian.