Ningaloo. Broome. Bungle Bungles. Not exactly world-famous names, are they? But then Western Australia doesn’t mind lacking big-name destinations, because it knows it’s got the best: a reef superior to the Great Barrier; wineries to rival the Hunter Valley; an outback wilder than Uluru’s; and sunset-spangled beaches that beat anything else on the continent.

Yet for all its attractions, the state gets just 13% of Australia’s visitors. That may have something to do with its size. Western Australia is vast — a full third of the nation’s landmass — and its highlights are scattered. Backpackers with months on their hands can pick them off in leisurely, land-based style, but for the rest of us, a trip needs to be carefully planned. Internal flights are a necessity, and they need to be booked in advance.

The good news is that with a bit of forethought, three weeks is just enough time to see the sights, get active and still set aside a few days to work on your tan. Here’s how.

Days 1-3: FREMANTLE

Perth is so nearly a gorgeous city: it has an ideal climate and a winning setting on the majestic Swan River. Somehow, though, it lacks soul — its centre is a bit too modern, a bit too businesslike, a bit too, well, dull.

A better base for your first day or two is the nearby suburb of Fremantle. This used to be Perth’s neglected port, until it got a comprehensive face-lift ready for the 1987 America’s Cup. Suddenly, “Freo” boomed. It’s now a buzzing little place with a youthful populace, a harbour full of yachts, and streets lined with jarrah-wood-accoutred coffee shops and bars.

Your flight is likely to arrive in the late afternoon or at night. Don’t hire a car; take a taxi (£23, 40min) to Freo and head straight out for dinner on the quayside. Little Creatures (00 61-8 9430 5555), with its on-site brewery, distils the mood of the place.

Next morning, stroll to the Old Fremantle Prison (8 9336 9200; £8) for a fascinating tour of Australia’s penitentiary past. Dating from the mid-19th century, the prison has hosted assorted colourful characters, some of whom make surprise guest appearances during your visit.

After an early lunch, take the train to Perth Central (£1.20, 25min) for a quick look around the city itself. King’s Park has the best views. Make sure you’re back on the train well before dusk, though, because two stops before Freo is Cottesloe, Perth’s premier beach. If it’s warm enough, go for a late-afternoon swim. If not, bask in your first sea-dipped Western Australian sunset — ideally with cocktail in hand. The Indiana Tea House (8 9385 5005) has a great selection and is good for dinner.

Idyllic sandy beaches and easy cycling make Rottnest Island a superb place to chill. Six miles by three and all but car-free, it is reminiscent of America’s Cape Cod — gentle hills, sandy dunes and exceptionally pretty bays. Spend day three here: it’s an hour from Freo by ferry (Rottnest Express: 8 9430 5844; £19 return; 9.30am and 11.30am, returns 4pm) or sailing yacht (Starsand: 8 9316 3591; £39; 8.30am, returns 4pm).

From the jetty at Thomson Bay, hire a bike (8 9292 5105; £8) and take off. It’s easy to loop the island in half a day, and the best beach stops are at Little Salmon Bay and Little Parakeet Bay. En route, watch out for the island’s sweet but vaguely scary quokkas: half kangaroo, half rat, they’re endemic here.

Back in Freo for dinner, check out the chilli prawns at Marconi’s (8 9335 3215) along South Terrace: sensational.

Days 4-5: MARGARET RIVER

Times recommends

• Dubai cheaper than Doncaster as room rates tumble
• World Cup 2010 - how to follow England
• Copenhagen: Europe's greenest city