Hi Daz
Suburbs very close to beach/CBD are: Bondi, Bondi Junction, Bronte, Clovelly, Randwick, Tamarama, Coogee. If you go to
whereis.com - Unsupported browser and search any of these you will see where it is in relation to the CBD. Bondi Junction has a train/tube station, the rest rely on buses. Don't bother with Double Bay area, it's very expensive and there's no proper beach there.
For rental properties, checkout
Real Estate, Property, Land and Homes for Sale, lease and rent - realestate.com.au and search rentals, nsw then these areas.
Hope that helps.