Yangmingshan National Park is less than 1 hour bus ride from Taipei and it's great for some hiking and nature-watching. You can also combine the hot springs in Beitou and do an amazing full day trip!
One of the easiest way to reach the park is taking the red line of the MRT until Jiantan and then the S15 bus. The last stop is Qingtiangang that is exactly where the trails start. There are various options and you need to go further into the wild in order to be immerse in nature without having people around.
I climbed up to Mt. Qixing (2100mt) that is the highest peak and it takes around 1.5 hour of stairs up to the main peak and then went down to Xiaoyoukeng passing through the volcano. A part from this trail there are really tons of different options if you want some hint shorter or less steep! Once you reach Qingtiangang you can decide which one to do and spend one day lost in the green :)
If you're doing my same itinerary then you can take the bus 108 from Xiaoyoukeng until Yangmingshan Bus Station and then one of the many buses to Taipei centre.
Here's what's left of my map! :)
Enjoy getting lost! :)