The mountain at the top of Taipei is not a secret, and it does not require a personality. On a cloudy weekday it is a public park with steam vents and a wind that comes off the sea. Take the bus from Jiantan (Yangmingshan National Park) or a taxi if you are four people and bad at schedules.
Silvergrass season is the postcard. The rest of the year is better, because you can hear yourself. Bring a layer even in September. Eat noodles in the village, not a concept. Be back before the afternoon rain if you are new to the trails; the mountain makes its own weather, and it is not the same weather as Zhongshan.
If you only do one hike in your first six months, do this one. Then go east.