My quick takes as an outsider (and I hope the locals will pardon my limited experience as a foreigner):
Taiwan:
I would say it seems to be a better place to live than it is necessarily to visit. Very Western in some parts. Some amazing parks along the east coast. In the cities, like Taipei, you can get food every 10 feet... and a cheap cab every 5 seconds. I could definitely pass on places like the Snake Market (aka Snake Alley), but the night markets are also where all the local kids come together and socialize... like the piazzas in Italy. The National Palace Museum is cool for all the stuff that used to be in Beijing's Forbidden City. And if you can get out to Tainan, I recommend it.
Hong Kong:
Everyone lives at least 14 stories up. Beautiful place if you can actually see through the sky over the harbor. Yet while the typical SE Asian major city is always hot, unbearably humid, and choked with air pollution, I'd say Hong Kong is one of the better ones (versus, say, a Manila or a Bangkok). Still, window air conditioners will drip all over you from several stories up in many parts of the city. Amazing food. I would have loved to have taken a day trip over to nearby Macau and probably will if/when I go back.
Singapore:
Oh, do these guys love their food. Lots of good Indian food too. The place has a completely exaggerated reputation for its Draconian laws -- the gum chewing, the caning, etc. Again, hot and humid as hell (which is one of the reasons I don't like SE Asia much and live in a place like SF). But most people escape from the heat because the place is like one gigantic shopping mall. And whatever you do, avoid lame places like Setanta Island. I found Singapore is better as an indoor, rather than outdoor, experience. I'm not sure I'd go back.
Beijing:
Wide swaths of construction. Brand new, cheap-looking high rises are cropping up everywhere like Magic Rocks, but there are a few parts where the architecture is a little less slapdash, as you might say in the UK. I found that many of the locals, like much of China, seems to have a major chip on its shoulder about an exaggerated inferiority complex, thinking everyone else is the world is out to put them down. So with preparations for things like the 2008 Olympics, they're going way over the top in trying to impress the world with their status and newfound riches.
Except in China, everything is always just a little bit off and incongruous with expectations from people in the rest of the world. The best way I can describe it is that it's like ordering a hamburger and 95% of it is exactly what you'd expect - meat, bun, condiments. And then there's that 5% - like a chicken foot thrown on top - where the locals don't see the oddity in things that others in the world might see. Whatever it is, it's always interesting because of those differences.
And the city "blocks" are massive. In the center, intersections can be 1-2km apart with traffic jams at every one and 80km/hr straightaways inbetween. And fake brand name merchandise everywhere. And people hawking DVDs for < $1. And be careful for Mongolian desert winds that cover the city with grit. And be wary of buying bottled water on the streets, as there's a lot of people refilling plastic bottles from questionable sources.
I highly recommend Tiantan and even the Summer Palace.