This mountain actually has its own MRT station (Xiangshan, which means Elephant Mountain). Just exit and follow the signs for about five minutes to the trailhead. Elephant Mountain is the vantage point of all the classic shots of Taipei 101, so expect a steep trail up. Weekends it gets crowded, especially around sunset. Don't forget to take water.
Various trails take you to different viewpoints; there are bilingual maps at key points.