Sandiaojiao Lighthouse — A lighthouse built in 1935 with panoramic views.
Dahu Park — 25-minute MRT from the center of Taipei. It has a large lake and is close to several hills and hiking trails.
Dajia Riverside Park — Great for bike rides along the river.
Xinsheng Park Area of Taipei Expo Park — The food court and farmer’s market happen on the weekends.
Day trip (<1 hour drive from the city) Jiufen — A beautiful town in the hills northeast of Taipei, famous for its food (particularly fish and meatball soup!) You'll get there within an hour by car, but it's 1:30mins by bus.
Beitou — An easy train ride out of the city center to Yangmingshan National Park with its Beitou district, famous for open-air hot springs.
Wulai — Natural springs, waterfalls, and extensive traditional delicacies served up by the local Atayan people.
Weekend trip (>1 hour drive from the city)
Kaohsiung City — Take the Highspeed Rail, train, or bus to visit the beach.
You'll notice Tai Chi being practiced in parks individually or in groups. There are a lot of schools where you can learn the practice, like the Inner Research Institute.
There is an active group of expats gathering regularly to play beach or grass volleyball, here's a map of their locations.