Historical Taipei: Chiang Kai Shek's Mountaintop Mansion, and the National Palace Museum
The first site I got to was in downtown Taipei-- a historic theater in the Taipei Ximending area-- I noticed it as we came out of the MRT to go to Ximending (which is a local night-life/young people area). Great history (most of it was in Chinese) and there are some local artists with their wares in the back. It was built when Taiwan was a Japanese colony-- originally as a market building, then a theater, and now a tourist attraction.
Yangmingshuwu (aka Zhongxing Guesthouse) was the summer residence for Chiang Kai-Shek in the mountains of Taipei. Full of historical photographs of meetings with important people (mostly Western powers) and relics from the Chiang's life (like old outfits and a watercolor stand)-- it was a pretty grand place (which made me wonder about how lavish their life must have been). Beautiful grounds with secret tunnels, alcoves for hidden guards and machine guns-- this is like Taipei's version of Hearst Castle. The easiest way to get here is probably by taxi-- and there are roughly 4 tours that run every day (probably best to call them before you go). It didn't seem like you were allowed to wander around the premises.
The National Palace Museum is probably most famous for it's cabbage and pork belly jade carvings. Unfortunately the cabbage has been relocated to the newly opened National Palace Museum to the south (but they have pictures of it up in case you wanted a closer look). Really intricate jade carvings, woodblock prints, furniture, sculptures, cauldrons, paintings, and jewelry are located here. Apparently only 1% of the collection is displayed at any one time. If you walk to the library (which is across from the main building and behind the food court area) there is a timeline that shows roughly when things were brought to Taiwan from China. Basically Chiang Kai Shek "stole" the artifacts from Beijing and brought them (mostly on the beds of pickup trucks) across rafts and into a vault that's been dug into the side of the mountain. Apparently Chiang not only stole the artifacts but also stole most of the gold, which is why Taiwan was a force to be reckoned with in the late 40s. Easiest way to get here: MRT then take a cab. There is a bus system that runs but it's quite a distance from the station and cabs are cheap.