First time in Cornwall, itinerary suggestions without a car

Hi all, I'm planning to go to Cornwall in early September with my wife and golden retriever, we live in London and are planning on getting the train down. We don't plan on hiring a car so looking for a suggestion of how to split a week.

I was looking at Falmouth as a starting point given ease of getting there and I can see there is some nice day trips such as St Mawes. I'd appreciate any other tips. Is September a good time to visit in terms of weather still being warm? Where else should we go? Should we base ourselves in Falmouth for entire week or do 3/4 days there and then go somewhere else for the rest of the time?

Thanks in advance