We're visiting Greece for the first time end of September to mid October. I don't like packing/unpacking. Athens is Base 2. Needing suggestions for base 1,
I’d fly directly to the island of Naxos after arrival at the Athens airport (give yourself 2+ hours connection time).
It’s a perfect island for an introduction to Greece. No cruise ship crowds. Very walkable, but also lots to see on the island with a car rental or bus or taxi tour. Great restaurants. Reasonably priced hotels. Plenty to do for a week or more. Easy transfer via plane or ferry back to Athens.
mlk35, just an FYI. If you want more responses, you might want to move this thread to the Greece forum. You have it in Greece Reviews, which is where people leave reviews for hotels and restaurants and so on. A lot of people here don't check this and so might miss it.
You also might consider changing the title so that folks understand what you're asking for. For example, instead of "Two Bases," title it something like "Need suggestion for base in Greece," or something like that.