I guess they want to take the train, so the mountains, as beautiful as they are, are not really an option without own vehicle or taking a bus. I think Guanwu has a tourist shuttle service, but still kind of cumbersome to get there.
When taking the HSR, you can go down to Kaohsiung too in less then two hours. Countless options, hence needs to narrow down the focus.
It’s a wooden train station built by the Japanese back in 1928.
At the station you could rent a YouBike, and ride to a near by Mangrove/crab trail (賞蟹步道). If you don’t want to ride north to the crab trail, you can also get off the train at Sanxingqiao station, rent your YouBike there, then ride south to the crab trail, then continue south to Xiangshan station.
I don’t know what you could do with your YouBike from there, you might have to ride it all the way back to Xiangshan. Or if you could get it on to the commuter train at Qiting station, you can return it once you are back in Hsinchu.
I don’t have a car. I can take a taxi or uber if it is possible. I don’t want to spend the night. And, I don’t want to spend more than 5-6 hours on the train/bus/taxi.
If you enjoy walking, walking from downtown Neiwan to Matuy trail (馬胎古道 is doable. You walk along the river to Yixing bridge (義興橋), then cross the bridge to get to the entrance of Matuy trail, then take Nanping trail (南坪古道) back to Neiwan suspension bridge and back to Neiwan.
It would be pretty much like this video
On your way to Matuy, if you walk pass Yixing bridge, you can get to Beijiao (北角), cross the Beijiao suspension bridge for some ice cream. It’s a 2.2 km walk from Nanwan suspension bridge. I think there is also the 5631 bus that could take you between the suspension bridges, but they only come every 2 hours.
On your way back from Nanping, you can take a soak at Venetian Bay Hotspring before crossing the Neiwan bridge.
Nice walk through lush forest, up to a small indigenous settlement (not much to see there), quite different from the crowded (weekends and holidays) old street area of Neiwan.
Luoshan forest road is a nice place to walk in the mountains if, for whatever reason, you can’t/don’t like to hike through steep trails and stairs. It is a concrete road and our idea was to drive through the south section and walk if convenient/safe. But at the point where north and south diverge https://goo.gl/maps/N6dWD8Z3m7RuXG2S9 , the concrete road for the south section disappears and becomes a trail. So, we drove a couple of km through the north section, parked and walked until km5. It was the long weekend in April, so there was a few cars and motorcycles, even people camped in some places, but overall quiet and safe to walk with a 3 years old.
I don’t know for how long it keeps being a concrete road, but judging for some motorcycles and cars that we saw going up and down, it should be quite a few km more.