You've just walked through Terminal One Customs and out into the passenger area, turning left to go to buses. Terminal Two is different (no pictures here.) Just go to Kwo Kuang Bus Line
There are about five bus companies going to Taipei Train Station. I've taken one of the buses on the left. We got stuck in local street traffic for 25 minutes before we got onto the highway. I've heard some people say their buses are newer. The bus I took was not good. The last times I've taken the bus on the right, Kwo Kuang, they were new. All are air conditioned with NOT air conditioned toilets. This bus always gets on the highway directly from the airport and makes only 1 stop before the train station. It's your call.
45 minutes if normal traffic.
Taiwan International Airport Terminal One bus counters
After buying ticket to Taipei Train Station stand in this line
Kwo Kuang Bus just dropped you off at Taipei Train Station EAST doors side (new drop off location.) Now walk up to the most friendly girl you see and ask her out to dinner. (About 3 seconds wait). Arriving girls this will work for you too.
Look to the left for HF2
Look for Hitachi sign on top of building as you get closer. Near Circle K "OK" store.
More pictures from the EAST side of the Taipei
Train Station. (Happy Family 2 is on the EAST side of the train station.).
This the WEST side of the train station in case you come from this side.