If you like national parks, hot springs/geysers, mountains, scenic places and natural paths...
A free option is having a leisurely hike up the rolling hillsides of Yangmingshan. The park is accessible by public bus on the outskirts of Taipei just like BeitouGeothermal Valley is.
It is a big national park with world-class scenery and facilities. There are even campgrounds.
Below approximately 1,000 meters (3,280 ft), the vegetation is subtropical rainforest, and above 1,000 meters, scrubby trees and tall clumping grasses are the main form of vegetation.
You can have a nice walk or go on long hikes. There are hot springs like Beitou, volcanic sulfur vents, and many nature trails to explore.
You can sightsee and take restful walks on the sightseeing observation points that were constructed near the sulfur fumaroles.
You can also go on long hikes. Yangmingshan National Park (Yang Ming Shan Guojia Gongyuan) is a collection of peaks on the northern outskirts of Taipei. The tallest peak, Mount Qixing (Seven Star Mountain), has an elevation of 1,120 meters (3,674 feet), and is considered the highest point in the municipal area of Taipei City.
Weather tip: The weather is fine all the year around, but when it rains some of the steeper trails are slippery.