Official sites

taiwan.net.tw is the official website of Taiwan Tourism Bureau.

Taiwan Tourism Bureau

From this website, you will learn most tourism information of Taiwan.

Taiwan Tourism Bureau

On the website you can choose the area by name

They are

Keelung City
Taipei County
Taipei City
Taoyuan County
Hsinchu County
Hsinchu City
Miaoli County
Taichung County
Taichung City
Changhua County
Nantou County
Yunlin County
Chiayi County
Chiayi City
Tainan County
Tainan City
Kaohsiung County
Kaohsiung City
Pingtung County
Yilan County
Hualien County
Taitung County    Taitung Government tourist site

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Penghu County
Kinmen County
Lienchiang County

But I believe for those you never been in Taiwan, those name does not mean anything to you.
Surly you can browse around the website and find someting interesting, but unless you have a tour guild or friend to guild you there, otherwise it is still impossible for you to go there alone.

For example you found "Little Liouciou" is very attractive, but on the website it does not provide too much information about how to go there and where to stay.

Taiwan Tourism Bureau

Taiwan Tourism Bureau