For Taipei long layovers here is a list of day trips accessible by train or bus. The train time table in English is easy to access
here.
Most of the good day trips are on the Taipei/Keelung line. The train is a slow commuter train, although for a longer trip to Hualien there are faster options. You can buy individual tickets for these trips or just use the Taiwan
Easy Card. I highly recommend getting an Easy Card if you come to Taiwan, it just makes life simple for riding the bus, MRT, and trains.
Fulong: A Fulong day trip takes about 6-8 hours. It is about 1:40 one way on the slow train (cost NTD83 one way). Fulong is actually a beach (there really aren't nice beaches in Taiwan, but this is about the best one in the north of the island). The activity here is a 27km bike trail that goes along the coast in a big loop. Bicycle rentals are right outside of the train station. The trail is well sign posted and fairly flat for riding. The bike rentals shops have reasonable 15-21 speed Giant hybrid bikes for NTD100/day. Take the road towards the 2km old train tunnel to start, that gets you out to the coast. On weekends there will be some small cafes and shops open along the way. Weekdays is a bit more hit and miss so better to take enough water with you for the entire trip. One offshoot of the ride is up to the lighthouse along the coast. You'll pass the steps up where the bike route says to turn right. If you stay on the highway at that point you'll see the road up to the lighthouse a few hundred meters down on the left side.
Sandiaoling: This is on the exact same train line and a little closer to Taipei. It is around 1:10 on the train from Taipei station (NTD59 one way). This is a hike of around 2-3 hours depending how far you want to go up the mountain (bring water with you from Taipei or go over the river to the small town to stock up before proceeding up the trail). From the small train station keep walking along the tracks away from Taipei. You'll come to a very small group of building on the right side before the bridge over the river. This is the trailhead. On the weekend they'll will be a fair amount of people, weekdays far fewer. This trail is fairly easy and generally well maintained. It has some rope bridges and even metal stairs on one steep section. There are 3 waterfalls along this trail. I've only ever gone up to the third waterfall and then back down. On the weekend there are some restaurants in the village over the river that serve a reasonable Chinese style lunch. If you did keep going after the third waterfall it is possible to go over the top and down the other side to another town and train station for a ride back to Taipei.
Jiufen - Jiufen is accessed via the Ruifang stop along the same rail line as the two above. It is a few stops short of Sandiaoling. This is more of a touristy stop that Flight Attendants would like
It's known for the narrow alleyways of its old town, packed with teahouses, street-food shacks and souvenir shops. It can get quite packed on the weekends with tourist busses. To get up to Jiufen from Ruifang there is a taxi stand outside of the train station. There is also a bus that goes up to the old town. Either way works.
Yangmingshan National Park - This is the little mountain north of Taipei city. There are dozens of hiking trails on the mountain. To get to the top of the highest mountain take a bus or taxi up to the National Park HQ. There are some smaller buses that do a loop of the scenic sites, but this is more about hiking. From the National Park HQ continue up the road about 100 meters to the trailhead for the Qixing Shan. Good guide
here. This is a moderately difficult hike but is paved most of the way up. There are no facilities beyond the trailhead so take plenty of water/food with you. From the top of Qixing Shan on a clear day there are great views of Taipei and also to the north coast. There are two other trails down from the top, one towards Lengshuikeng and another towards Xiaoyoukeng. I prefer the former as there are more options for getting back down the mountain and it's a little bit more scenic. From Lengshuikeng visitor center there are two buses that go back toward the Yangmingshan bus terminal where you can change buses back down to Taipei or grab a taxi back to the city.
There are many more day trips, but this list is a good start.