I have many ABT friends who would write about their experiences in Taiwan much better than I, but it nonetheless is one of my favorite places on earth. It's so relaxed, kind, cheap, and just polite. In particular, Stephanie writes great guides about Taipei in The Thousandth Girl.
My favorite neighborhood in Taipei is actually Dongmen 东门, as I find the stores and boutiques very quaint around there. For classic Taiwanese, 大隱酒食 James Kitchen is great in that neighborhood. Climb Xiangshan 象山 for the best view of Taipei 101!
Take a day trip to Jiufen九分. If you leave around 4, then you get to watch the gorgeous sunset AND see the famous restaurant that Spirited Away is based on at night. From MRT Zhongxiao Fuxing (忠孝復興), Brown Line and Blue Line interchange, leave from Exit 3 and take bus route 1062 (Keelung Bus) to Jinguashi, alight at Jiufen. The ride is about 1 hour and the fare is NT$102.
TCRC: THE EARL GREY PASSION IS MY FAVORITE DRINK IN THE WORLD. I repeat, in all caps, THE EARLY GREY PASSION IS MY FAVORITE DRINK IN THE WORLD. But, everything in this bar just blows every bar out of the water.
后面有个小酒吧: ...Except for the old-fashioned here. The meticulousness of the bartender carmelizing sugar and bitters on the top of a grapefruit to soak in the drink is incredible.
小滿食堂: As my friend sam puts it, coziness is comfort to the extreme, which is exactly what this place is. A really well put together concept, with home-cooking and even homier service.
Gone: Because the bar owners run a little hostel upstairs, their concept of hospitality is right on point. Delicious cocktails, and plus you get to sit underneath the cloud lights!
TAITUNG > HUALIEN
Why does no one talk about how Taiwan looks like Hawaii? Crystal clear water on one side and roaring mountains on the other. It'a about 3-4 hours to motorbike from Taitung to Hualien, and the drive is so awe-inspiring. Most of it is right next to the ocean, so you get to feel the salt and spray, but there's also stretches towards the end through farmland and banana groves, winding mountain roads and steep climbs. You could it by bike as well, but I'm lazy.
If you can time it correctly, sunset over 8 arch bridge is such a #nonfilter experience.
IMO, Hualien looks exactly a little Tainan, and there's a pretty touristy little night market, but it has amazing crazy vegetables!!
Also Zhenggong is a super famous baozi store that you should stop by and snack on!
Taroko Gorge is beautiful but I wouldn't spent more than half a day there. Definitely don't ride your motorbike in, if you're not experienced, as there are some tight turns and narrows roads with incoming traffic. In my opinion, motorbike is definitely the best way of getting all the majesty of the gorge-- without the windows of a bus or car. There are some interesting hikes on the original trails made by the Japanese to fight the indigenous peoples-- stop at the super helpful visitor center for a map!