EDIT: ugh, wall of text.. I was too excited to point out the awesome that is Taiwan, and I had to cut out several paragraphs because there's so many things you can do in Taiwan. I'm not even Taiwanese...
Please say that you'll be closer to Taipei, than be in Taoyuan.. . Well, ok Taoyuan is not that bad, but Taipei is infinitely more fun. ..Taoyuan = zzzzz.....
There are gyms that cost around... ~100NT (~3USD) per hour or two (but they never really check after the first hour). My girlfriend took me there, but I'll have to ask her where exactly they are. I can get back to you when I found out.
Taipei is modern, and cheap. Man, after being there for so long, I don't think 1 month is long enough. However, try to avoid shady motels that also have 2-hour rates. A lot of them are actually very clean, but still... every hotel room has condoms with 2-4 channels of porn haha.
My favorite night market in Taipei was shutdown recently... w/ more likely to follow suit to follow the new rigid health laws. Sigh, so sad.. my first date w/ the gf was there too. My absolute favorite nightmarket is in Keelung (Chi'lung). It's a cheap 55NT ride to Keelung from Taipei bus station (right next to the Taipei main station). Try not go to there on a weekend there.. It's so packed.
Oh, and tell the Pakistani dude who makes delicious meat wraps (heh, they're essentially burritos) that "Daniel from Meiguo says Hi." You can also tell that bum-artist dude in next to the nightmarket in the main underground crosswalk that "Daniel from Meiguo says hi." Avoid the train tracks there though.. UGH...
If you have a chance, I'd highly recommend hitting up Jiufen:
http://wikitravel.org/en/Jiufen
and Hualien:
http://wikitravel.org/en/Hualien (and the national park nearby, Taroko).
Hualien has this one mochi shop which has uber-delicious mochi snacks. The tinytiny nightmarket in Hualien has this ultra-delicious "Coffin Bread." It's essentially two fried slices of bread with a covered hole on top that looks like a coffin.. which covers the delicious meat in it.
I took trains/bus all around that country, and had an absolute blast. I practically lived there for about a third of my life in the past 4 years (due to work). Public transportation is so easy there, and getting around and talking to anyone wasn't an issue. Taiwan probably has the most density of English speakers of any asian country I've been to (except Philippines, and Singapore ahah).
Damn.. if you only have a month, then I'll probably recommend you take the train that goes along the eastern seaboard, and essentially go b&b/hostile hopping down towards Kaohsiung and Kenting.
If you want to get to know a bunch of young US expats, I recommend heading to "Playing Lounge Bar." in Taipei. My close buddy is a regular there, and if you want, I can PM you on the details of that bar. It's essentially a hookah lounge.
http://www.taiwannights.com/en/bar-lounge/playing-lounge-bar-taipei-135
From there, sometimes we'd go to a club afterwards where foreigners get in for free (well, only on weekends). Yay for me!
The Taiwanese people are possibly the most down-to-earth, genuinely nice people on Earth. I'm not kidding. I've touched every single goddamn continent on this earth (well, I almost touched antartica.. I didn't disembark ), and I have to say Taiwanese people are the friendliest people in the world. I met my cute gf there, and I am absolutely stoked that I'll be there again for work this summer (Kaohsiung).
Oh, and if you're in Kaohsiung, be sure to visit "Bottoms Up" next to the Grand Hilai hotel (tallest hotel in Kaohsiung, hard to miss). It's the most unlikeliest place to find the best steak in Taiwan. A US expat from New York owns that place, and he's a great dude to chat up. On my first encounter with that bar, I passed out in their toilet until 3-4am.. my shipmate thought I ditched him, but he ditched me :'(
efff.. you can probably tell that I miss that place. It's my favorite country to visit in the world(with New Zealand being a close 2nd...). My taiwanese buddies back home in the US tell me that I know more about their country than them haha