Are there any islands close to Santorini to stay?

Ferry schedules don't always allow for daytrips. The closest island is Ios, then Naxos.

ferry schedules: www.gtp.gr

Anafi is much closer than Naxos and is supposed to be a beautiful place (although I've never been).

Ios is located less than 50 minutes by ferry from Santorini, so you can definitely visit it after a few days on the volcanic island.

If you are considering island hopping... Some suggestions for your Greek island hopping around Santorini would be these:
Anafi, Folegandros, Naxos, Ios, Mykonos. You might also consider: Tinos, Paros, Amorgos, Crete.

Ios is also connected via several routes by ferry or boat to Paros and Naxos. If you start your journey from the further destination, Naxos is closest to Santorini and Ios, so you should visit it first before going to Paros and then back to Piraeus. If you want to make the opposite, visit first Paros. Just remember the route: Piraeus-Paros-Naxos-Ios-Santorini. Amorgos is connected only with Naxos or Paros by ferry. So, from Santorini or Ios, you have to go to Amorgos via Paros or Naxos. Mykonos is connected with Naxos, Ios and Santorini. So, to go from Mykonos to Paros or Amorgos, you have to do that via Naxos.

It depends on what kind of vacation you are looking for. There is a Greek island for every taste ;)

Your question can be taken 2 ways, and it seems that most people think you want to stay on Santorini and do day trips to another island. However, I think you mean the opposite .. AND in fact you posted an earlier question to that effect, https://community.ricksteves.com/travel-forum/greece/island-hopping-advice-for-two-families-and-two-weeks-of-travel-time and I and "Stanbr" told you about the excursion company that operates out of Naxos & Paros

To give full info on This company, here's the link -- https://visit-cyclades.com/day-cruises-naxos-cruises/ -- It operates 2 vessels do day excursions to Santorini, MV Alexander on Tuesdays, Naxos Star on Wednesdays. I recommend the former, because it's larger & much smoother ride. These boats leave Naxos 7:45 , pick up passengers at Paros & reach Santorini 10:45. A bus takes u from the port to OIA, there's a guide but u have about 2-3 hours free time there, then bus takes you to main town Fira, ditto, finally you re-board to go down to port around 5:15, back in Naxos between 7:30-8pm. Cost has gone up; now €65 per adult, €35 per child 5-11. However, when u consider that the basic economy ferry ticket Naxos - Santorini is now over €20, with bus/taxi $$ on top of that, and it's a fair deal. Enough time to enjoy the Famous Clifftop View, and the upward look upon arrival/departure, check out the Very worthwhile museum etc ... and be back on your home island in time for a peaceful sunset... worth consideration.

As far as choosing between Paros and Naxos, both have sandy beaches (which Santorini does not) and even in High season, accommodation rates are much more do-able than Santorini. This forum has a large contingent of Naxophiles, including me, but I get tired of repeating myself. RS doesn't make it easy to search back threads, but I found one with a description by me, abnd also a post by wonderful "stanbr" who includes links to his FAB photos -- here it is -- https://community.ricksteves.com/travel-forum/greece/mykonos-and-naxos-first-time

I must warn you that you are VERY Late in booking accommodations for the Popular Cycladic islands in High Season -- jillions of Europeans have got there before you and snapped up the best budget lodgings closest to the beaches. To save you time in unrewarding searches, I'll just say ... if you folks want a "villa" that will house you all, then you can't be all that near a beach, because all the beach areas are taken up with small hotels & pensions... free-standing "villas" are a relatively recent phenomenon in these 2 islands, and thus must be inland a way, so you'd have to rent a car for your stay. You'll have better luck if you lok for apartments or what are called "1 br studio units" -- lodgings that have a br, pllus large main room w. kitchenette, 2 daybeds, table/chairs etc + balcony. On Naxos, Ag. Giorgios beach adjacent to the port town, is handiest, (walk everywhere -- only rent car for exploring around the island), and good for young ones, but gets crowded in High season. Ag. Prokopios is v. large beach plenty of room to spread out... has cafes, etc. Ag. Anna IMHO is a too-narrow too-crowded beach... Plaka has pllenty of room even in high season, is farther out. Good bus service however, to all. Good luck hunting!