We visited St Augustine earlier this year and stayed at the Bayfront Marin House B&B. Nice place on the main drag, easy walking distance to shops & restaurants.
St. Augustine Hotel :: #1 Rated Hotel on TripAdvisor
While you did mention a trip in Sept or Oct if possible consider going during the Nights of Lights time frame Nov 19th to Jan 31st. Beautiful to walk around and see all of those lights after a great day of sight seeing.
St. Augustine Nights of Lights | 2016 - 2017
I do not think anybody has yet mentioned it but our favorite spot was visiting the Castillo de San Marcos National Monument. Just amazing to see that old fort and listen to some of the speakers there.