Take a suburban train from Central Station (track 24) to Bondi Junction and then bus 378 towards Bronte Beach. Get off at the terminus. The travel takes 30 minutes from Central. See  for more info.
Bronte Beach homepage!
There is very limited shopping in Bronte such as convenience stores.
However, there is world class shopping in nearby Bondi Junction.
Cafés & Restaurants - line the street down at the beach.
There are no hotels in Bronte but accommodation can be found in private B&Bs and holiday apartments.
The nearest hotels are in Coogee.