One of my favorite natural places within reasonable distance from Taipei City is Full Moon Gardens (Man Yue Yuan). The closest town is San Xia, and depending on how you go, is about 40-50 minutes drive from Taipei City. There are buses that go there a few times a day, but I wouldn’t rely on them.
Full Moon Gardens has tons of safe and reasonable trails, rivers, streams, waterfalls, trees, and occasional animals, if you’re quiet enough to not scare them away. It’s not too crowded, depending on when you go. If you are really adventurous, you can hike for many hours through the extended trail system.
This location is thoroughly written about in Richard Saunder’s book Taipei Day Trips 2, available through the Community Center in Tienmu (community.com.tw)
By the way, when I posted about this place on Facebook, my friend PM’d me and asked me to not post about it, cause it was his “best kept secret place”
Well worth the trip!! Enjoy!