Catch the Ferry to Manly, then the Route 199 bus to Palm Beach (1 hour) and do the walk up to the lighthouse 30 minutes each way.

https://www.nationalparks.nsw.gov.au...oey-lighthouse

Catch the train to Wentworth Falls in the Blue Mountains and do the 20km bushwalk to Katoomba

https://www.nationalparks.nsw.gov.au...cliff-top-walk