We stayed on the outskirts of Fira. If we returned, we'd stay in Imerovigli.
Do the Fira to Oia hike and then bus back. Look up https://santorinidave.com/ for a detailed map of the route.
Check out the open air cinema in Kamari. I've never been to a cinema as good before or since.
We did hire a car and it was a game changer to be able to explore easily, and at any hour so I can't comment on the buses. The only time we got the bus was at the end of the hike as we'd parked in Fira. Santorini is a small island compared to the UK, but its not small. Even driving takes about an hour from end to end.
Pyrgos is beautiful and unspoiled.
The views from the monastery are incredible. You can actually see the shape of the island, and the dots of white that are Fira and Oia. You can also see planes below you, coming in to land at the airport.
The airport by the way is little bigger than a shed. Departures consists of a small kiosk and some plastic chairs.
Do a boat tour out to the volcano. We took one that also included the thermal springs. You jump from the boat and then swim to the springs which are at the shore.
The queue for the cable car back up from the port was over an hour long so we opted to walk back up. Keep your wits about you as you will play Dodge The Donkey as they come thundering down the stairs.
We went right at the beginning of October and it was perfect. It was still beautiful sunshine and warm, but not unbearably hot, and it wasn't too busy. Just enough people that it felt like it had atmosphere. Different parts of the island have their own microclimates. We were having breakfast in Kamari and were a bit chilly. Weather apps told us that it was 26° in Pyrgos, so we went there.