Tamsui is at the end of Taipei's Red Line. The Old Street has a variety of restaurants and shops and you can reach the oceanfront from here, in contrast to the bulk of Taipei which is several miles inland.
The Chiang Kai-Shek Memorial Hall is one of Taipei's most distinctive landmarks. The museum on the bottom levels paints an over-the-top picture of Chiang Kai-Shek's life. The 228 Memorial Hall a few blocks away is worth a look for a more balanced take on the excesses of the Kuomintang.
The Chungking Mansions are considered by many to be the closest thing in Hong Kong today to the Kowloon Walled City. While they are a far cry from the density of the Walled City, the bustling interior corridors have a similarly seedy reputation with Hong Kongers.
The Southern District of Hong Kong island is now easily reachable with the newly opened South Island MTR line. Visit Aberdeen in the evening and try the fish ball soup served directly from one of the fishing boats returning to the harbour for the night with their catch.
The Chi Lin Nunnery and Nan Lian Garden are located in Kowloon and are open to the public. The Maitreya Hall uses traditional Chinese buidling techniques and was constructed without the use of any nails.