My favourite fact about Taiwan's capital, Taipei, is this: within two years no resident nor office worker will be further than 50 metres from an entrance to an underground station. It puts the estimate that Londoners are never more than 20 yards from a rat to shame.
Taiwan has been off the map, almost literally, for decades from the British traveller's perspective. It used to style itself the "Republic of China", much to the irritation of the much larger People's Republic of China, 100 miles away on the mainland – which has long regarded this surprising and dramatic island as a thorn in its ideological side.
China exacted all kinds of diplomatic revenge: when BA flew briefly to Taipei, for example, the airline had to create a subsidiary, "British Asia Airways", to serve the route for fear of losing its valuable rights to fly to Beijing.
But rapprochement across the Taiwan Strait means that the first non-stop flights from London start next weekend, helping to open up an island that has much to offer besides excellent public transport. Early in the year of the tiger, Taiwan is burning bright.
In 1971, I led a pioneering tour to the People's Republic of China; at the height of the "Cultural Revolution", we were treated to an abundance of propaganda. This month, I redressed the ideological balance with a visit to Taiwan – and, for the necessary history lesson, headed straight to Taipei's memorial to Chiang Kai-Shek.
You can't miss it: modelled on the temples in the Forbidden City in Beijing, it continues a Chinese tradition of honouring former rulers with folie de grandeur rather than grandeur on its own and is therefore plumb in the centre of the city. Carefully tendered lawns and flower beds on all sides, and then the marble of the building itself, topped with blue-glazed tiles, make clear that this is a shrine and nothing else. Eighty-seven steps – one for each year of Chiang's life – take pilgrims from ground level to the exhibition hall.
Chiang Kai-Shek was the ruler of mainland China from the late 1920s until he was driven out by Mao Tse-tung in 1949, after which he had to make do with governing Taiwan until his death in 1975, although he always dreamed about returning "home".
Political anoraks well versed in the precarious geopolitics in this part of the Eastern world will spot the distorted view of Chinese history that Chiang's memorial represents, but watching school parties being taken around shows modern Taiwan at its best. They are the first generation of Chinese children to hear public ridicule from their teachers at former leaders, and to be able to laugh at grandiose portraits where once their parents had to show reverence. Guides point out, too, the life-size model built of Chiang's study. It has six doors, so concerned was he that he should have scope for escape should one or two of the entrances be used by potential assassins.
Human life was of little concern to Chiang Kai-Shek – but Chinese art certainly was. Towns might be surrendered, armies slaughtered and the currency devalued, but the treasures of the Beijing Palace Museum were carefully packed in the 1930s, carried around China in the 1940s and then brought to Taiwan in 1949. The current museum first opened in 1965 and even with subsequent expansion, only a small part of the collection can be viewed at any one time. Follow the crowds and the noise to see the major attractions: jade, porcelain and scrolls.
Taiwan's headlong rush towards the future is manifest in the Taipei 101 skyscraper, which is named after its number of storeys and possesses some of the best elevators in the world. Upon arrival, visitors are likely to feel frustrated that they are kept stationary in the lift while an attendant reels off statistics in three languages, but they will forgive her 45 seconds later when they realise that it has already risen 1,200ft to the 89th floor, without anyone noticing. As you gaze down on an urban sprawl that melts into the haze, you notice two prominent bridges, side by side. So enamoured was the Nationalist regime with the US general Douglas MacArthur that he gets not one, but two bridges named after him: MacArthur Bridge One and MacArthur Bridge Two.
In a country founded on its embrace of capitalism and hatred of communism, it is perhaps surprising to find such an addiction to public transport . Besides the rapidly expanding underground system, Taipei also has a reliable fleet of buses. Take them to get from A to B in air-conditioned comfort – and also for the company. A non-Chinese face is still a rarity on board. If you so much as glance at a map or guidebook you will instantly be offered help. And if all you do is peer out of the window, you will look in vain for a smoker on the pavement, a political poster or a drunk, or litter.
Going underground is the most sensible policy when you are hungry: the best food is usually to be found in the subterranean food courts where a number of restaurants share a common table area, so diners can choose selections from each. This saves time – and service charges. Menus are illustrated and usually translated into English, which means you can, if you wish, avoid delicacies such as fatty pork steamed sandwich or pig's blood cakes. Bean curd, on the other hand, crops up everywhere, as do spring onions. Taiwan does produce a sweet red wine, but the leading European gift to Taiwan is in the shape of the excellent locally produced beer – conveniently called "Taiwan".
The first Europeans to conquer Taiwan (the island, not the beer) were the Dutch, who arrived in the 17th century and strengthened their hand with an array of sturdy forts, which all subsequent conquerors could put to use. When you leave the capital on the high-speed train line that runs with spectacular efficiency all along the west coast, an excellent first stop is Tainan. Two hours down the track, this city is still strewn with colonial relics, including a fort that still has its original Dutch cannon.
Rather than fortifications, the British established consulates instead. These were enclaves, rather like little colonies, where they could ignore the Chinese authorities and operate their own legal system. Some of these are still functioning. The consulate in Kaohsiung, a little further south from Tainan, is a splendidly isolated red-brick fortress on a peninsula just outside the town. British tourists who clamber up the steep steps to get there are welcomed on arrival with a Beatles exhibition and recordings of "Come Together".
Considering 20 million people are crammed into an island less than twice the size of Wales, Taiwan also boasts some prodigious wildernesses. The largest of these, Kenting National Park, stretches across the southern tip of the island from the west to the east coast. It has one or two conventional seaside resorts, but the needs of its birds (300 species) and butterflies (200 species) always take priority. And in Kenting National Park you find more proof that, while other nations talk green, Taiwan gets on with low-impact transport.
The People's Republic was the land of the bicycle for decades, but on recent visits I have been alarmed by a sudden love affair with the car – causing traffic jams in towns and cities that are choking in at least two senses. Across the strait in Taiwan, the government still champions two human-powered wheels.
Indeed, to encourage cycling the country has adopted the concept of a drive-in hotel. One at the southern end of the island has an elevated check-in desk to save cyclists from the hassle of dismounting. They then ride directly into the lift and on to their rooms, which are all equipped with bicycle racks. There is even a bike spa, for those that need repairs, or a wash.
After a week in Taiwan, I felt I had seen China as I dreamed it could be, full of bicycles, narrow-gauge railways, contemplative monks and teenagers happy to come to terms with its troubled history. The future begins here.
Travel essentials: Taiwan
* The author flew with EVA Air (020-7380 8300; evaair.com), which has daily flights from Heathrow to Taipei via Bangkok. Fares are currently £585 return in economy and £885 in "Elite" class, which is premium economy.
* EVA will be joined on the route by China Airlines (020-7436 9001; china-airlines.com) from 28 March, flying non-stop from Heathrow to Taipei three times a week.
* Yoho Bike Hotel, Kenting, Ping Tung County (00 886 8 886 9999; yohobikehotel.com.tw). Doubles start at NT$7,300 (£151), room only.
* Formosan Naruwan Hotel and Resort, Taitung (00 886 89 239 666; naruwan-hotel.com.tw). Doubles start at NT$2,000 (£41), room only.
* Taiwan Tourist Board: taiwan.net.tw