Here are some of the hikes I've done in the Taipei area - and some that are a bit far out but can still be done from Taipei in a day.
Let's start with my Most Recommended hikes:
Xiaotzukeng Old Trail (Pingxi area to Jiufen) - a fun, not too challenging hike that includes a ridge walk (with little view) but as it gets to the top, you step quite suddenly into an opening that gives a heartstopping view of the ocean, Keelung Mountain, Jiufen and Jinguashi, and descends to Jiufen.
Wankengtou Mountain and Caoling Old Trail (Dali to Daxi, Yilan County) - a long ascent on a historic trail (you can drive this part actually, a road winds up along the old trail) and then an up-and down ridge walk with phenomenal views over the Pacific on one side and rolling grassy mountain peaks on the other. It's very long - leave early.
Yuemeikeng Waterfall (Jiaoxi, Yilan County) - the most beautiful waterfall you'll find in Taiwan. It requires some river tracing but it's not a long hike and definitely worth it. In fact, I'll go so far as to say that if you live in Taipei for any length of time you have to do this hike.
Pingdeng Canal Old Trail (Yangming Mountain) - good views at the end from Qingtiangang
Sandiaoling (Pingxi area - a longish day hike) - not much on the stunning vistas but lots of lovely nature
A Taipei City Hike (Jiantan to Neihu - and it really is right in nature!) - this starts near MRT Jiantan station, climbs above the temples near it, peaks a few mountains in an unsettled part of northern Taipei, offers a view of the National Palace museum, gets farther into nature than you ever thought possible in Taipei and ends up at the top of Little Gold Face Mountain (小金面山) with rocky outcroppings and a stunning view over Taipei.
Battleship Rock (scroll down to where the photos and directions are)
Longdong Ridge Hike, Rock Scramble and Swimming Hole
Some other great hikes:
Dutiful Son Mountain (Pingxi, blogged by my husband)
A Maokong Hike (Maokong)
Paozilun Waterfall (Shenkeng - easy peasy short hike)
Shulongjian and an uncompleted hike to Stone Bamboo Shoot Mountain (Jingtong, near Pingxi)
Dajian Mountain (Xizhi)
Elephant Mountain (Taipei City) - great views over Taipei/Xinyi
Xiao Wulai (not "Wulai" itself - this barely counts as a full-on hike and it's hard to get to - leave super early - but still worth it)
Shengxing to the Railroad Bridge (Miaoli County - again, hardly a "hike" but worth it and can be done in a day from Taipei)
Heping Island Pseudo-hike (Keelung) - a huge chunk of this is like a real hike, on a hillside trail above the ocean with some cool rock formations and temples
Easy Taipei Day Hikes For Late Sleepers
Dongshige Old Trail (Pingxi)
Other hikes I've been on but not blogged about, because I did them before starting this blog:
Bitou Cape - very easy hike (hardly a "hike" at all) on Taiwan's north coast
Laomei Waterfall - easy day hike from Taipei but leave early enough that you don't get stuck on the road with no buses at night, very close to the northernmost tip of Taiwan. Your best bet is to take a bus to Laomei town from Danshui and to ask directions in the town for the turn-off to the road that eventually leads to the trail - or buy Taipei Day Trips and follow the directions.
Tianmu Old Trail - start from the upper level by Wenhua (Culture) University on Yangming Mountain, go on a weekday early enough or late in the afternoon and see monkeys! This one is easy to get to - any bus to Culture University on Yangming Mountain will deposit you near the top (walk towards Culture University/the big old-looking round Chinese building, turn right at the Catholic church in another Chinese-style building, and follow the trail leading down to the left, or start from the very end of Zhongshan N. Road - keep going north past the traffic circle until you reach the base of the mountain, and start walking up the stairs).
Seven Star Mountain and Xiaoyoukeng Sulphur Spring - a classic Taipei day hike on Yangmingshan
Jingmei Mountain / Xianjiyan - I will probably post about this soonish even though I did it a long time ago. This hill near MRT Jingmei will take you up, across the spot where Lu Dongbin (the deity enshrined in Maokong's Zhinan Temple) stopped to search for his lost lover and deposit you in Muzha.
Wufengchi Waterfalls - hardly counts as a hike, it's more a series of safe, easy stairs to three pretty waterfalls above the hot springs town of Jiaoxi in Yilan County.