You can’t take a train to Sanxia but it’s only NT100 from the Yingge train station to the famous Zushi Temple in Sansia. You can take a bus but it’s hard to know where to get off.
Other good short train rides: take the train to Ruifang and transfer to the Pingxi small rail line (my favorite). Good stops include Sandiaoling for the waterfall hike; Shifen (for the coal museum and Shifen waterfall - the broadest fall in Taiwan); Pingxi to light off a sky lantern or hike in the hills just across the main road (interesting craggy peaks to scale, small caves to explore); and the final stop Jingtong. Some good hiking here, the remains of an old mining settlement, and a cool old station to explore.
Sanyi also makes a good day trip if you catch the express train. Explore the wood carving shops, the wood carving museum, and walk along 12km of abandoned tracks through pretty countryside. Also some good Hakka restaurants around.
Fulong and Daxi, in addition to the beaches are great for the Caoling Historic Trail. You can walk up to 15km though shorter routes are possible. Fantastic views of the coastline as well as a chance to learn about native plants. There are signs everywhere (in English and Chinese) explaining the local fauna. The great thing is when you finish at Daxi you can go for a swim and then take the train back. No need to retrace your steps.
Hsinchu surprisingly makes for a good day trip by train. The Guqifeng museum in a large temple complex has a fabulous collection of Chinese antiquities. The temple grounds are also filled with hundreds of good quality statues. In town there are a number of relics to explore, including the excellently preserved south gate. There is also a cool movie theatre from the 1920’s where you can watch modern films for NT20. Oh yeah, the glass museum is worth a visit and you can arrange English speaking tours.
Oh yeah, Neiwan. There was a lot of work being done when I was there in February. Building new trails, pavilions, etc. If nothing else, take Almas Johns advise and grab a few beer and head down to the river for a swim.
Neiwan is also good for those who want to try Hakka foods, expecially lei cha (pounded tea). But eat or drink in town near the train station. The restaurants across the river served bland coffee-shop style food and shitty tea. At least the two I tried.