The Royal Botanic Gardens is a lush oasis of greenery that lies in the fabric of the city of Sydney. It is at the edge of Sydney Harbour near or surrounding landmarks like the Sydney Opera House, the Art Gallery of New South Wales, and Parliament House. The Domain is a popular recreation area just south of the Gardens, and it could be rationalized that it is an extension of them. While the Royal Botanic Gardens is the result of an organization founded to cultivate plants from Australia and elsewhere, the Domain is more of a city park utilized by locals as their collective front lawn.
The Royal Botanic Gardens includes the National Herbarium of New South Wales, the Sydney Tropical Centre, the Rose Garden, and the Palm Grove Centre. The latter area features a visitor center, the Gardens Shop, dining areas and public toilets. Free guided walks of the Royal Botanic Gardens start at the visitor center every morning. A "trackless train" tour operates in good weather conditions, but this tour costs a few dollars. Otherwise, feel free to wander aimlessly and enjoy a bit of nature within the urban jungle. Look around you to see the skyscrapers peering over the treetops. A walk through here is quite refreshing after a shower, as you may get the feel of a rainforest in a tropical area. Besides the multiplicity of flora, you may spot a strange bird or two tiptoeing on the lawns.
The Royal Botanic Gardens is accessible during daylight hours throughout the year. Unless there is some ticketed event, a visit here is free to most of the areas. The Domain is open all the time, though I am not sure you would want to saunter through here in the middle of the night.
Try to walk all the way to Mrs. Macquaries Point, which juts out into the water and offers some splendid views of Sydney. Look for the old rock formation nicknamed Mrs. Macquaries Chair, a spot named after the wife of the former governor who enjoyed this view but had fewer landmarks to gaze at back in the old days.