I am a Chinese tutor who loves to hike. The time I spend on hiking is even more than teaching.
I have been hiking around Taipei for many years, yet I can't finish all the trails. Wherever you see a hill here, there's almost always a trail to get to.
I personally prefer Ruifang area the most. When the weather is fine, the view on the ridge is amazing. You can see the beautiful coast with blue sky and sea. On a cloudy day you might see the sea of cloud on a little mountain. These mountains are not high (maybe only a few hundred meters ), but the views are incredible.
Pingxi and YangMingShan are also fun places. There are a lot of trails you can go by public transportation. The trails are like spider webs. If you don't have a map it is easily to get lost. Since they link together, you can decide how far you want to go. It can be a couple of hours or up to 12 hours. Some trails are more challenging than others.
I can customize the hike for you according to your need.