I only had 2 days so didn't venture out of the city but I would love to see the more rural areas!
The highlight for me was the big museum, I forget what it's called but it's like the biggest museum in the world. The Taipei 101 is also worth a trip. I've been underwhelmed with city views elsewhere after seeing this one.
We took cabs everywhere since they were cheap, but the metro is supposed to be amazing and would cost virtually nothing. I found it overall to be bet affordable.
We stayed at Beauty Hotel - Roumei Boutique, which was great but not sure if the location is optimal. Everything seems pretty spread out so I don't know if there's 1 central area you want to stay in. The hotel had a futuristic, awesome toilet and jacuzzi with a TV in the bathroom wall. There were 2 channels in English, CNN International and porn, LOL!
I haven't been to Taipei, but I live close and I've thought about going many times so I've done some research.
For a day trip I'd go to Beitou and visit the hot springs. There is also a nightmarket outside the city a bit, by the coast, that seems worth the trip. I do t remember what it's called by Bourdain went there on the Taipei episode of The Layover.
I'd also get lots of foots massages. And I think having your hair washed is a thing there? I might be getting confused with somewhere else though.
And of course, eat as many xiao long bao as humanly possible.