Santorini 3 nights and where else for 3 nights? Naxos, Ios, Paros?

As you plan, keep in mind that Santorini is too far south from other islands for day trips. Paros and Naxos are near each other and have nice enough beaches. Santorini is not known for it’s beaches, rather for it’s caldera views. We loved staying on Paros and were happy to leave Santorini.
Hotels will be most expensive on Santorini.
You can go out on a boat there.
Your hotel budget is going to be a challenge.

Sometimes you can find a ferry timetable that will allow for a daytrip to another nearby island. Using the www.gtp.gr website I arbitrarily chose Saturday, May 7 to look for a daytrip. I found a departure from Santorini at 10:05 AM for Naxos and a return to Santorini leaving Naxos at 11:05 PM. That gives you around 12 hours to explore. Check on your proposed travel date to see what's available to you then.

After time in Athens, Meteora, and the RS Greece Tour, I flew to Santorini for 3 nights, enjoyed the island including a day-long island tour with Akrotiri ruins, and another day inland on my own. The caldera coast was very nice, but very touristy and often crowded. Then I took a ferry, about a 2.5 hour trip, to Naxos for 3 nights, which I loved. Much more Greek community life. Fewer tourists, in large part because Naxos has no cruise port, just a ferry dock and a small airport. I also arranged for a day with a private guide with car, to tour inland towns and sites. Then an early morning flight back to Athens for one last day, before flying home the next morning.

Paros or Naxos is fine. Don't go to Ios, unless you are into crazy parties with heavy drinking or even drug use.

My choice would be Naxos and then Santorini. As mentioned earlier, there is no cruise port in Naxos, hence less tourists on shore excursions. We hired a private guide here and he was terrific in helping us explore the island. In my experience it will be easier to be frugal on Naxos than on Santorini. I can recommend Lithies Traditional Homes in Santorini. It's in Firostefani with a beautiful view of the caldera. Maria will provide you with exceptional service. We found it to be the best value for our money in Santorini if you want to stay on the caldera. Worth every penny! https://www.lithies.com/

To change from the usual recommendations (Naxos & Paros) , look at Amorgos, the island of the famous film "The Big Blue".
You will have everything you are looking for, beach, boats, snorkeling, diving, restaurants, hiking, the Monastery of Panagia Chozoviotissa, traditional village with windmills (in Chora), less the crowds of Santorini.

https://www.visitgreece.gr/islands/cyclades/amorgos/

In Aegiali one of the two ports of Amorgos there is for example a Spa hotel overlooking the sea called "Aegialis Hotel&Spa"
https://amorgos-aegialis.com/
(there are other accommodation options at all prices and especially cheaper than in Santorini)

Amorgos is about 1h15 from Santorini by daily Seajets ferry connection.
You can rent a car when arriving at the port of Katapola or take the bus or request a transfer to the hotel.

To return to Athens you can leave from Aegiali port or Katapola port.
The crossing takes between 7 and 9 hours depending on the ferry