The Cairns War Memorial stands at the top of the Esplanade near the junction with Shields Street, and the surrounding civic precinct contains the best of Cairns' early 20th-century architecture. The former Post Office, the Supreme Court, and the School of Arts building — now the Cairns Museum — were all built between the 1900s and 1930s, in a style that combines federation era solidity with tropical adaptations: deep verandahs, louvred shutters, and covered walkways.
The Cairns Museum, housed in the School of Arts building (built 1907), covers the history of the region from Aboriginal occupation through the gold rush era, the Chinese community, and sugar agriculture. It's small but well-curated and gives essential context for understanding the city. The admission is modest and it's worth an hour.
Walking from here south along Abbott Street to the art gallery and courthouse complex gives you a decent survey of central Cairns architecture in about 20 minutes.