Taipei and environs: 
- National Palace Museum. Take the bus from Shilin or Jiantan, or, if you're feeling adventurous, walk from Jiantan (it's over a scenic mountain that passes by a monastery with a good view of the city and a butterfly sanctuary which probably isn't open in November) The seasonal exhibits in November are paintings and calligraphy, and the slightly longer-term exhibit is about literati life in the 17th century and Warring States period jade
- Jiufen, aka basically Spirited Away town. I've not been and heard it's a bit of a tourist trap, but you seemed interested when I brought it up
- Wulai is basically the fastest way you can experience aboriginal Taiwan culture being a 1hr bus trip from Xindian subway station up the mountain. The area has hot springs and waterfalls, a hilariously small gauge train, and aboriginal cuisine (and museum). I liked the sausages, and they use a distinctive spice called maqaw
- Night market, pretty much any one of them. Shilin night market is the most popular, but has too many people for my tastes. You're here to eat whatever you can get your hands on, usually all sorts of interesting street food (last time I went I had barbeque octopus tentacle, which was interesting). I would recommend seeing if you can get some Scholar Cake (狀元糕), even if it's just to watch it being made
- Oh yeah, go find a maimai machine and play it. I think you can find arcades in the basement mall sections that you can walk from Taipei Main Station MRT
- Food: most of everything is amazing. Get some Taiwanese beef broth noodle soup. Suggestion for hole-in-the-wall with high quality: 桃源街老王記牛肉麵店 (it's a little pricy as this dish goes but the broth is very rich and the beef is high quality). Like honestly, go into a lot of side streets in vaguely touristy areas and there are tons of good eateries
- Funnily enough, the one cuisine that is really hard to find right in Taiwan is Cantonese cuisine, so eat up *that* while you're in HK
- Oh yeah Busses in Taipei: Go get an EasyCard (think Oyster card etc.), which can be used on busses has the added benefit of being able to use them at convenience stores. Check the sign above the driver, it will say if you need to pay on getting on or when you alight (上車付 vs 下車付), and if it changes while you are on the bus you are expected to pay again