Kaohsiung is amazing. The nightlife is the best in TW (except Taipei obviously) and the weather is better. The beaches are great too! Kaohsiung also has a pretty decent MRT/Bus system - not as good as Taipei obviously but serviceable.
I've never been to Hsinchu other than to see the historic temple market-y thing. It seemed alright I guess, I dunno, a bit more gloomy than Kaohsiung. It's hard to say but IMO Kaohsiung/Taichung are better.
Taichung is pretty cool, the museum is good and...I dunno really it's like Birmingham in the UK, it's in the middle, and there are a lot of people, but nothing really seems to happen. There's obviously a bit of a problem with the nightlife there too, I'd steer clear. (Oh, and from what I remember, the public transport was crap).
I'd say Kaohsiung, the people are the friendliest I've met in Taiwan and it's got a slightly more Polynesian feel to it than the rest of Taiwan, which makes a change. But it's up to you really. For me, the only places I'd live in Taiwan are Taipei or Kaohsiung, but it's subjective.