Not to miss

Royal Botanic Gardens

These expansive gardens are the city's favourite picnic spot, jogging route and strolling venue. Bordering Farm Cove, east of the Sydney Opera House, the enchanting gardens were established in 1816 and feature plant life from the South Pacific and around the world. They include the site of the colony's first paltry vegetable patch. Long before the convicts arrived this was an initiation ground for the Cadigal people.

The Sydney Tropical Centre is housed in the interconnecting Arc and Pyramid glasshouses. It's a great place to visit on a cool day as it's always warm. The multistorey Arc has climbers and trailers from the world's rainforests, while the Pyramid houses Australian natives, including monsoonal and tropical rainforest plants. Other attractions in the gardens include the Fernery and the Rose Garden.

Free guided walks (10:30 Mon-Sun and 13:00 Mon-Fri) depart from the information booth at the Gardens Shop, and last about an hour and a half. As far as wildlife goes, you can't fail to notice the gardens' colony of grey-headed flying foxes, or the large flocks of sulphur-crested cockatoos.

The Gardens' paths are mostly wheelchair accessible, though there are some flights of stairs about. Attractions are well signposted, although the estimated walking times are best described as pessimistic.

Sydney Opera House

Gloriously white and brilliantly sharp, Australia's most recognisable icon sits dramatically at the tip of Bennelong Point. On a sunny day the Opera House is postcard-perfect; its startling curves and points a pinnacle of architectural genius inspired by the segments of a mandarin. It's a truly memorable place to see a performance, listen to a free outdoor concert or sit under a cafe umbrella and watch harbour life go by.

The Opera House is so unique that it has been photographed a zillion times, appears on an army of cheap T-shirts (and every other Sydney postcard) and decorates the frames of Dame Edna's dramatic glasses. The construction was itself truly operatic - it was even dramatised as The Eighth Wonder, performed here by Opera Australia in 1995. The interiors don't live up to the promise of the dazzling exterior, but if you're curious to see inside there's an informative, hour-long tour (9250 7250; Guided Tours Office, Lower Concourse; adult/concession around 26/around 18; ;09:00-17:00); not all tours can visit all theatres because of rehearsals, but you're more likely to see everything if you go early.

It was built between 1959 and 1973, but was plagued with construction delays and political difficulties culminating in the resignation of architect Jørn Utzon in 1966. The interior was designed by a consortium of Australians after Utzon quit.

Art Gallery of NSW

With its elegant Classical Greek frontage and modern rear end, the state's main gallery has an outstanding permanent display of Australian art (including a substantial Aboriginal and Torres Strait Islander collection), a well-regarded Asian gallery, a Western collection starting from the 16th century, and some inspired temporary exhibits (prices vary).

The frequently controversial, much-debated Archibald Prize exhibition is held here every March, with portraits of the famous and not-so-famous bringing out the art critic in every Sydneysider. There are free guided tours leaving on the hour from 11:00 to 14:00 (Tue-Sun), and wheelchair access is good. Look out for Lin Onus' Fruit Bats (1991), a cheeky bunch of critters painted in Aboriginal cross-hatching hanging from a clothes line.