Most foreigners in northern Taiwan who don't live in Taipei City have to make a trip to the Taipei County Foreign Affairs Police Station (map D-3; the station is marked by a circled red X). Here are diections for how to do so using public transportation.
To get to the police station, first go to the Far Eastern Department Store (Yuandong baihuo) (map: D-3, the red dot on the border with D-4). The store is known as FE21.
By MRT and bus to FE21:
Get off at Hsinpu (Xinpu), the final stop (map: A-4). Leave the station by exit 1 (advantage: additional bus that is free) or exit 5 (advantages: shorter walk to busstop, better chance of getting a seat). If you go out exit 1, walk to the spot marked "B" on the "Hsinpu" map; from there you can catch a free trolley-style bus from the department story or a 245 public bus (see note below on 245 buses). If you go out of the station by exit 5, walk to the spot marked "G" on the "Hsinpu" map; from there you can catch a 245 bus (see below).
By bus from Taipei to FE21:
I don't recommend this relatively slow way, but if you want you can take a 245 bus (see below) from the Taipei New Park (2-28 Memorial Park), which is south of the Taipei Train Station.
Once you're at FE21, follow the road along its right side. The road will curve to the right. When the road ends, go left. The entrance to the police station is just past the fire station. Again, the police station is marked by a circled red X in area D-3 of the Banqiao map.
The "center blue 245" bus (with 245 in blue in the center of the sign on the front of the bus) 245 and "red 245 bus" 245 are the fastest ways to get to FE21 -- and, consequently, the police station -- from the MRT. Be careful not to confuse the "center blue" bus with the "left blue" bus.
Do not take the "green 245 bus" 245, because it takes the long way and people unfamiliar with the neighborhood can easily get disoriented on it.
The "left blue 245" bus 245 goes straight down Wenhua Road, turns left at the intersection at the bottom right corner of C2, and then turns right on Zhongshan Road. You should get off just after the bus turns onto Zhongshan, then walk back along the road to the department store. But you're probably better off catching either the "center blue" or "red" 245 bus.