Between May and September, buses connect Mykonos Town to many (but not all) of the beaches. For the most popular beaches they depart from a terminal called South Station (more commonly known as “Fabrika”), but there's another terminal by the Old Port, often referred to as “North Station” (see the Mykonos beaches map to locate these bus terminals). There’s no bus connecting these terminals, you have to walk 15 to 20 minutes through the streets of Chora.
The fare is between €1 and €3, depending on the distance traveled, and you can buy tickets directly from the driver or at the booths at the terminals.
Some buses from the most popular beaches run until around 3AM in July and August (as is the case of Paradise), but in most cases there are just around half a dozen buses throughout the day. Departure times are clearly marked at the terminals, and vary every year and even throughout the season, so always double-check before departing, especially the return times.
In the summer of 2019, the beaches with the best bus connections are Platys Gialos (from Fabrika -- every half hour -- and the Old Port -- every 90 minutes), Paradise (from Fabrika -- every 30 minutes, at 15 and 45 past the hour during the day, and then hourly, at 15 past the hour from 7pm), Ornos and Agios Ioannis (both from both Fabrika and the Old Port, but more frequently from Fabrika -- every half hour).