Doing that Touristy Thing in Taipei: Taipei 101 and Tamsui
Taipei 101 is probably the most well known building in Taipei itself. It was at one point the world's tallest building. The mass damper (that big sphere thing) is what absorbs the energy from the wind that hits the building. It is the largest damper sphere in the world. There are two observatory decks-- inside and outside, but frankly the inside one has a lot more to see-- I mean, there were tons of outdoor tracks and helicopter landing pads and things you could see from the inside deck! You can also watch the planes fly in and out of the nearby airport.
Tamsui is a small fishing town located at the end of the red line from Taipei. It seems mostly like a touristy spot for young families and couples-- and has a sort of Venice Beach/SF Fisherman's Wharf feel to it. There's a pretty famous bridge known as "Lover's Bridge" which is about a 45min ride by bus from the MRT station. Decently crowded on a Monday night. Decent Italian food can be had if you walk from the MRT by the water past most of the shops....