Freycinet National Park
Freycinet National Park is on the East Coast of Tasmania. It is one of Tasmania's first National Parks, being established in 1916. There are several very nice beaches there, including the famous Wineglass Bay.
Freycinet National Park is administered by the Parks and Wildlife Service of Tasmania. http://www.parks.tas.gov.au/natparks/freycinet/
Flora and fauna
It's about two and a half hours drive from Hobart.
You need a permit for all of Tasmania's National Parks. They can be bought at the national park visitor centre at the park entrance, and there are several different options from a day pass for just this one national park to a pass that is valid for two months and includes all National Parks of Tasmania.