From Sydney take a train on the Northern or North Shore Line from Central, Town Hall or Wynyard. Get off at North Sydney and change to any bus headed for Neutral Bay (175, 178, 184, 202, 203, 207-209, 227-230, 247).
Once on Military Road, alight at the first stop and stay there. Look out for the 143 or 144 bus which will take you over Spit Bridge to Manly Wharf in approx 20 minutes. Total travel from the city centre is around 40 minutes.