i would say the things to do are (in no particular order)
walk around Taroko Gorge and swim in one of the water holes around, stay in the formosa regent there.
sun moon lake , row across the lake. beautiful natural blue/green waters
national palace museum, top ten pct of the best of chinese art taken from China
yangmingshan , take a sulfur bath or two or three eat some local food like 3 cup chicken, then drive down to the coast and have some seafood at yehliu.
chihpen royal hotel - great hotel to relax in and take a nice sulfur bath after the beautiful drive along the coast from hualien.
drive from taipei to hualien along the coast
kenting and oluanpi , swim in the beautiful waters there
green island and orchid island (iv not been there but all the reports are positive bout them)
shihlin night market and the keelung night market for that taiwanese experience
cks memorial, sun yat sen memorial, taipei 101 and other Taipei city sights
Bear in mind that (and I realize I’ll probably get assaulted for this), unless you’re returning to your country of birth or to see your girl/boyfriend or have some other such reason, other countries have a lot more to offer to tourists. Taiwan is extremely densely populated, the buildings are ugly, and it’s a modern, fairly developed (ie., boring) country. Yes, there are interesting things to see here. And no that wasn’t intended as Taiwan bashing. But Vietnam, Thailand, Laos, Cambodia, Japan and the Phillipines, for example, are all more interesting.
I say that based in part on two experiences:
having planned sight-seeing itineraries for a friend, a brother, and my mother, on each of their visits, and
having just returned from Hanoi, where I saw so many more tourists than one sees here and I realized the reason was obvious: that country has so much more to offer.
But, given that, Taiwan’s still a nice place to visit. Depending on your time, I’d suggest:
Natl Palace Museum
CKS Memorial
Taipei 101, if you’re into giant phallic structures
A few temples
night market
Taroko Gorge and Hualien
snake alley, if you’re into gruesome butchering of snakes and turtles
stroll around a decent neighborhood in the morning/evening and watch the Taichi/dancing in the park
The nice Ju Ming sculpture garden/museum overlooking the ocean, followed by a good seafood meal on the coast
some good dumplings, perhaps at Din Tai Fung
That great food street, whose name I can’t remember presently, nearby DTF and Alleycats
I’ve just finished reading, “1000 Places to See Before You Die” by Patricia Schultz (a somewhat famous travel writer). It’s a library book. DON’T buy it.
Nada Zip Zero in Taiwan.
Not that I agree with her.
She’s into “seeing” expensive hotels.
She recommends “The Sukhothai” (hotel) in Bangkok, but passes over Sukhothai (the
ancient capital).
She says the Shanghai Museum is worth seeing, but not the NPM.