Shifen Waterfall is one of the most famous in Taiwan. It is not very tall (20m) but it is wide (40m) and powerful. It forms a horseshoe and has been nicknamed the Little Niagara even though it is much smaller. Shifen is a very popular stop along the historic and stunning Pingxi Railroad. The government recently took control of the waterfall area and it should be free now.
The hike: It is a 15 minute walk east from the town/station to the waterfall with plenty of signs (some in Chinese) along the way. You cannot drive to the waterfall. Alternatively you can do the longer Sandiaoling Waterfall Trail and get off the train in Sandiaoling. It will probably take about 3-4 hours to complete the trail including some time at Shifen Falls.
How to get there: From Taipei Main Station take the train to Ruifeng (1 hour) and then transfer to the Pingxi Line and get off at Shifen. It is also possible to take buses from the Muzha MRT station (link).
Other info: Supposed to be free now that it is under government control.
GPS info: 25.04887,121.78793