What areas in Greece to stay?

You may get a reply if you provide us with a bit of information. Something like when you are planning this trip, how long do you have? You mentioned 4 locations, does that mean you are interested in these 4 locations and want advice about them specifically. You can help us help you, only when you give us details.

I'm so sorry for not being very explicit, but I greatly appreciate your response. We are going to Santorini, Naxos, and Mykonos and wondering what area to stay that would be the most concentrated with shops, restaurants, etc. We are not really into the beech scene or swimming, so the water isn't really important.

Ok that is still pretty skimpy information. I will do my best to provide advice but suggestions can be different if you are going in high season versus shoulder season.

Experienced travellers to Greece often suggest you do Athens last. By doing islands first you get over jet lag and have a relaxing visit
and gives you time to get acclimatized to how things work in Greece. It is important to be on the mainland the night before your return flight.

I would plan to fly to Santorini first. It is amazing. We look for a hotel with Cladera view. The main village is Fira. It is crowded particularly when multiple cruise ships are arriving on the same day. Oia too is very popular and is very busy after the tour buses arrive between 9 and 10 AM. You can find nice caldera views in Firostefani just a 20 minute stroll along the sea view pathway into Fira.

3 nights is all my budget can afford because Santorini is expensive.

The ferry trip to Naxos takes just about a 1.5 hour. Naxos is an excellent companion island with Santorini. You will get a more realistic Greek experience in Naxos. For first time visitors I would suggest you consider a hotel at St George beach. It is part of Naxos town which is just a 10 minute stroll along the headland into the main waterfront promenade. You can enjoy all that Naxos town offers and have a quiet evening at St George. I know you said you are not particularly interested in beaches but you do not have to be swimming to enjoy St George ambience.
Naxos town is the transportation hub so you can get buses up into the mountains to visit quaint villages.

You should plan 5 nights in Naxos. You will not see it all but you will have a good experience. A week to 10 days is even better.

I have to be truthful, Mykonos is as expensive and hyped up as Santorini but does not deliver the same visual experience.
Frankly if you want to visit a great ancient site you could do a day excursion by boat from Naxos to visit Delos and then on to Mykonos for three hours. That way yo do not have to pay Mykonos prices to visit the island.

We always fly back to Athens from our last island. Athens needs at least a minimum of 3 nights to take in the ancient sites.
Some images
Santorini https://www.flickr.com/photos/stanbr54/albums/72157632154800782
Naxos town 2024 https://www.flickr.com/photos/stanbr54/albums/72177720318203165/
Naxos at night https://www.flickr.com/photos/stanbr54/albums/72177720318406428/
Trip around Naxos https://www.flickr.com/photos/stanbr54/albums/72157634605629689

Hope this helps