Italian Islands

Look at Rome2Rio.com. Sicily has several island destinations as well. You may have to do some backtracking to the mainland, or fly between some of them.

I've visited several Italian islands, but in separate trips.
Don't be fooled by how small they might appear on a globe, but the distances are great and unless you have months at your disposal, you won't be able to visit all the islands you mentioned in one trip.
If you are interested in visiting islands, you should concentrate on one big one or maybe a group of smaller islands located in close proximity to one another.
These are the groups you could consider:
- Tuscan Archipelago (Elba, Giglio, Pianosa, and maybe Montecristo, which is however a natural reserve accessible only with special permit)
- Sardinia (and maybe Maddalena Archipelago).
- Corsica
- Sicily (and maybe one archipelago close by, like the Aeolian islands)
Sardinia, Corsica and Sicily are huge and require at least 7-10 days each to visit at least to some degree.
- Gulf of Naples islands (Capri, Procida, Ischia)
- Ponziane islands (Ponza, Ventotene). These are sufficiently close to Naples to be combined with the Gulf of Naples if you have at least 2 weeks at your disposal.
There are direct connections between major islands (Sicily to/from Sardinia, Corsica to/from Sardinia, Corsica to/from Elba. Otherwise there may be connections between islands within the same group (for example the Aeolian island are connected to one another). But if for example you want to travel from one Archipelago to another, you might need to get back and connect through a port in the mainland. For example to go to Capri to Ponza you need to go back to Naples.