This mountain has its own MRT station (Xiangshan, which means Elephant Mountain). The trailhead is about an eight-minute walk south. The vantage point of many classic shots of Taipei 101, it's a steep climb 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.
Elephant Mountain is one of the Four Beasts; if you are feeling energetic you can hike the others (Tiger, Lion, and Leopard mountains) in about three hours. Just follow the signs.