First, a big thank-you for all the folks who were so helpful with my inquiry, a couple of weeks ago, about Santorini v. Naxos for a whole month. (Yes, we really do want to stay at least several weeks in one place.) We are going with the strong consensus -- unanimity, basically -- for Naxos, and we've chosen the Plaza Beach Hotel on Plaka Beach (any feedback about that one?). But, of course we must also visit Santorini, because I've never been there to see the bucket-list sunset and my wife, who was there twenty years ago, loved it. So we'll stay on Naxos for about four weeks and then go to Santorini for two or three days to end the trip (that's right -- an entire month in Greece without being a tourist in Athens!). What would be a good spluge-type hotel (under 300 $ or E -- almost the same now!), where we could get a good caldera view from our room? Ditto restaurant with good view and good view. Finally, "back" to Naxos: is there any reason to rent a car for the whole month, or just on as as-needed basis? (For many days on Naxos we'll probably be doing nothing except lying on the beach and strolling to tavernas.) And if you have any recommendations for little-known restaurants, sights, hikes, or events, please lay those on me. Thanks so much!
Have a look at the Santorini Palace. The caldera and sunset views are superb, as is the hotel. I could go on and on but look at the website and the photos. There was a bus stop just a short walk down the hill, and if you rent a car there is even parking space in front of the hotel. Each room has its own balcony, by the way. At sunset people from other hotels gathered close to the Palace to see the sunset.
Giving yourselves time to relax and settle in allows you to plan your excursions so you can just rent a car as needed. And from your posting I don't know if you were thinking of taking a car on the ferry from one island to the next, but that would not be allowed anyway. Be sure to go to your local AAA office to pick up an International Driving Permit, no test or membership required. Bring 2 passport photos to save a few dollars.
As for Santorini hotels with a caldera view, room rates depend on when you plan to be there, which you haven't mentioned. But with a $300/day budget in mind your best chance to find what you want would be to use a search website with a filter system for price, amenities and facilities, one that allows you to search village by village to narrow down your choices to a manageable level. I recommend www.booking.com for this. It's very helpful to weed out the ones that don't fit your needs. Villages with a caldera view are Akrotiri, Fira, Firostefani, Imerovigli and Oia. Fira is the island's social/commercial hub. Oia has the most beautiful and timeless architecture. Imerovigli has the best overall caldera and sunset views without the crowds. Firostefani is right next to Fira but doesn't attract all the cruise shippers that come ashore below Fira. Akrotiri is away from almost all the crowds except of course for the archaeological site there, so if you want remote that would work.
Don't shortchange Santorini with only 2 or 3 days! It's an amazing island with lots to see and do.
Try the Zoe Aegeas (www.booking.com) in Oia (it's a cave house). We stayed there 3 nights in April 2016 and had a wonderful experience with fabulous caldera view! Host Zak was great as was part-year resident/neighbor Colin.