Districts and neighborhoods in Taipei (or Taichung)

I’m looking for some advice. I have been teaching in Korea in the greater Seoul area for the past 2 years, and I also spent a few months traveling in Central America and the Philippines. Now I plan to come to Taiwan in about a month to find some work, but I’ve never actually been there before. My plan is to arrive on a visitor visa and hangout for a few weeks so I can check out some districts, schools, and neighborhoods in Taipei or even Taichung. However, a few weeks really isn’t all that much time, so I am looking for some help to steer me in the right direction.
Basically, I’m looking to live and work in an area that is:
• in or very near to the city (preferably close to a good nightlife and some interesting culture—restaurants, outdoor/night markets, contemporary art galleries, shopping, etc)
• convenient to get around by public transportation
• relatively affluent or recently developed.
• Near a good city park, or river way (with moderate or less pollution)
• clean and affordable housing (I realize affordable is relative, but you get the idea)
I’m used to city and “city-suburban” life, and I hope to find some work on the side in some nice neighborhoods. Some people have suggested places like Da’an, Neihu, Shilin, Muzha, in Taipei.
I guess I’m looking for a list of districts, areas, MRT stations or neighborhoods I should checkout in Taipei or possibly Taichung.
Also, if you have been to Seoul, here is a short list of areas I enjoyed which might give you an idea of what I’m looking for:
In the city: Hongdae, Gangnam, Apgujang , Silim, City Hall, Dongdaemun, and even parts of Itaewon (off the main street) were decent.
Outside the city: Bundang (best place I have ever lived…as far as convenience, cleanliness, and ability to make money…and only 15-30 minutes to downtown Seoul on the bus. I’ll miss it.)

I was thinking Ximen (a.k.a. Ximending, Wanhua district) or the area around Heping East Road, roughly in between Roosevelt Road and Da An Park. I limit myself to these two because you want affordable housing, but they’re nice places to live anyway, IMO. You’ll have a lot of public transportation there too (bus, mrt).

On Heping East Road you’ll never have to wait for buses for long. You’re near the MRT red line and brown line. Bus takes you directly to Anhe Road where there are many pubs. Da An Park is nearby. There’s a nightmarket nearby and there’s Yongkang Street. Clubs 'n pubs nearby. Brown line to Zhongxiao Fuxing shopping area… and where the blue line is; just two stops. Near the Shi Da university and the Shi Da language center… It’s a nice area.

I lived in Guting before (right on Roosevelt Road) which was very convenient but noisy, however the best place I stayed at was in Linguang. It’s basically “the end” of the big, ugly city but still with good access due to the MRT (brown line) and bus stop (many busses start/teminate at Linguang). Mountains at the back separate it from Muzha and give you that “green” feeling. There is a park nearby (Fuzhou “Mountain”).
It’s not too far from Anhe Road either that Ectoplasma mentioned, you can actually walk to there. Ditto for Taipei 101 / Xinyi shopping district.

Google Maps Link to Linguang