If you wish to travel from Fira to the main island, you can catch a bus run by the KTEL transportation company. In the peak season, you will find buses running frequently, although the schedules are subject to change. For travelling to Trissasia, you can catch a ferry from Ammoudi in Oia village or from the main port of Athinios.
Travelling in Santorini by bus is convenient but they get awfully crowded during the summer so you can catch a taxi if you wish to travel around Thirassia and Thira. The taxis generally have fixed rates but if you are travelling to an offbeat area you should agree to a price first.