I just planned my first Greek islands trip, going solo, leaving in a month. There used to be a small cruise ship company that did something like what you are looking for, would dock in the morning and depart late at night, so you'd have a full day on each island. They went out of business about a decade ago. I looked for a while, but only found large ships with little time in port - certainly not enough for dinners on shore.
I'm going to Crete, Santorini, and Naxos and taking ferries from one to another. There are at least 2-3 outfits that offer day trips from Naxos to Delos/Mykonos (about 3 hours on each), though they all seem to stop in September.
If you want to go on your own, it's certainly doable. This is the site to get ferry times and prices. If you are planning a lot of stops, it means a lot of planning, booking a room in each place, packing and unpacking every day or two, unless you have a lot of time for longer stays on fewer islands. Take into account that you probably need to be at the pier with ticket in hand (you can buy them on arrival to save time waiting in line) about an hour ahead, especially if you want to choose a good seat, and it takes a while to disembark - lots of passengers!! For long ferry rides (near the Turkish coast for example) consider flying. I compared Naxos-Athens by ferry and by plane, the plane was more convenient for me and the price was about the same.