Santorini Plus Island Hopping

The answer to what you are looking for is to rent a private boat with a skipper.

Depending on the type of boat: sailboat, catamaran yacht, its comfort, the services offered (cooking, cleaning), the time of year, it can be quite expensive. (more than 1000€/ day)
See an example here:
https://skippair.com/en/cruise/offers/private-sailing-cruise-santorini/#

I don't see any other solution to what you are trying to do.
The ferries in the Greek islands are only means of transport between the islands for the inhabitants, the vehicles and the goods, as are the buses or the trains in other places.

Taking a trip to the Greek islands is not like jumping from one ride to another in an amusement park or jumping from one tourist site to another in a big city. Each island needs several days to be visited and appreciated.

And Santorini is not ideal for island hopping as it is farther away from the other Cycladic Islands than most of the others.
As mentioned above, you have described a pricey alternative, a private cruise.

This is really helpful information. You're right -- we don't want to spend the amount of money needed for a private boat. It sounds like for a 2 week trip we should just plan on seeing 2 or 3 islands which is fine. I really think one of them should be Santorini. Can anyone recommend another one or two? I guess what I'm really asking is, what do you think a good 2-week itinerary would be for us?

To elaborate on what JoLui wrote, the main town on an island is usually where a ferry docks. If you spend a day there, all you'll have time for is either to explore that one town with maybe a bit of beach time, or take a guided bus tour of the island. Some islands are closer than others. For instance it's only an hour by ferry from Naxos to Mykonos or Paros, a bit more to Ios. The ferry is a short walk from the centre of Naxos. I've only been to Naxos, so I can't tell you if the ports on Paros or Ios are any different. I think Mykonos is, based on some guide book reading. Get a good guide book of the Greek islands to give you an idea of what there is to see and do and how much that interests you. I did go to Santorini and there is quite a bit to see there. You need to allow at least an hour to get from the ferry to your hotel and close to that to get back to the ferry. There will be hundreds of people waiting to get on/off the ferry.

Take a look at Crete. It's the largest island and has lots to see and do.

I can recommend Crete. However it is a large island so you will need to pick an area of it to stay in and explore that area. You can’t realistically explore things all over the island from one hotel. We had a car and three different hotels in 12 nights.
There’s a ferry Santorini to Heraklion Crete that only takes 2 hours or a bit less. Heraklion has some great ancient sights. But my favorite city was Chania, which is a few hours from Heraklion. We also stayed in a very small town near Matala on the south coast which was beautiful.

I can also recommend Nafplio which is not an island but a seafront city on the Pelopponese. Very pleasant city, great sights within easy day trip reach, beaches, everything.

From/to Santorini, the easiest routes would be Crete, a big island, to the point I think you forget you are on an island, we enjoyed it immensely, visiting places others mentioned. You could also go North into the Cyclades, most ferry routes stop at one or more of Ios, Paros, Naxos, and Mykonos. Not familiar with Ios, but Naxos is nice, Mykonos is popular, we enjoyed it, but like Santorini, highly touristed, Paros gets high marks from people. We also spent a week on a small island Koufonisia, it was great, but really beach is the attraction and it is a bit of a chore to get to.
I believe from Santorini you can also get over into the Western Cyclades, Folegandros, then on to Milos or Sifnos, all popular.

The key is to keep for the most part to the same ferry route if doing three. For example if you do not do Crete, then after Santorini stick either to the Western Cyclades, or going North, try not to mix.

Renting a chartered boat was mentioned, but there is a middle ground, there are small cruise ships...100 to 200 passengers, that will hit 3-4 Islands. Not sure of the cost, probably not cheap, but not private boat expensive. Windstar Cruises and Star Clippers are two, can't really make any recommendation of either, just seen them around.