First stop: Wulai. About a 1 hour bus ride south of Taipei, deep in the mountains, there is a beautiful aboriginal town, famous for hot springs and a waterfall. There is a cable car going to the top of the waterfall, where there is a huge resort. Although it was a bit far away, the scenery was refreshingly exotic.
Next stop: Keelung. There was an attempt to explore the NE coast of Taiwan, but pouring rain stopped that, so we settled for some of the best seafood in Taiwan in downtown Keelung near the harbor on the Northeast.
Next stop: Pingxi. Another small place deep within the mountains, yet a very short distance from Taipei City. Pingxi is where the Lantern Festival is, where people write their wishes for the New Year on a lantern and set it adrift. Again, the rain put a damper on a lot potential exploring.
Next stop: Wanli. More specifically, Yehliu Geopark. Wind, rain, and ocean waves can do some serious damage over time in terms of erosion, and we saw some of the crazy rock shapes that have been in the works for a long time (like "the Queen Head"). Another place I would have enjoyed more had it not been freezing and pouring, but a trip to the steaming Xinbeitou hot springs took care of potential frostbite.
Final stop: a mountain trip through Shijr on my trusty scooter. As crappy as I sometimes make where I live sound, a 15 minute scooter ride in any direction (except towards Taipei) brings you to a beautiful mountain road where you can give your lungs a bath and forget what apartment buildings look like. On the only decent day of the whole vacation, I had coffee on a peak with a 360 view of the mountain ranges of northern Taiwan.
Basically, rain sucks and Taiwan is a cool place, given the appropriate weather and transportation.