Places of interest / things to do on TAiwan for me:
North Taiwan:
TAipei city and immediate area= Taipei 101, Hsimenting , Tamshui , Pitan, Peitou, National Palace Museum. Taipei City Zoo, largest zoo in Asia. Take the cable car ride, cheap and good views.
Day trips from TAipei : Train to Fulung, swim in Fulung beach, train or bus back. Take in Keelung as a small side trip, see the little harbor and enjoy the little night market.
Visit the Kavalan whiskey factory in Ilan , its making a splash for itself. Take the train there. Spend the night in the nearby hot spring town of ChiaoShi. Visit the biggest fishing village in Taiwan (Nan Fan Ao, reachable by train and a short cab ride from Suao) to get some nice seafood at reasonable prices.
Visit Wulai , take the little RopeWay up there , take in some aboriginal food, maybe a hot spring at one of the hotels there.
Visit Yangmingshan, reachable in parts by bus, if you have a car its best. Then you can take in some hotsprings, either rustic or take in hotspring resort TIEN LAI.
TAke a train to the little town south of Taipei called CHONG LI, and take a one hour bus ride to SHIHMEN DAM. The reservoir behind the dam, they have a nice one hour boat ride thats cheap and nice.
TAICHUNG - central Taiwan :
Visit by bus/car the eastern part of the EW highway ending at Ku Kuan. There are a few nice hot spring hotels there. One small one is really nice, forget the name, but you walk over this small suspension bridge to get there. Great cabbage and fruits grown in the area, plus you may be able to eat the pond raised (near extinct) formosa trout. The EW highway is broken beyond KuKuan and not passable.
SunMoonLake. I Liked visiting the lake and there are few hotels around from ok priced to very expensive. I Liked boating on the lake. IT seems its crowded now with people from China, but should still be fun. The blue/green waters of SunMoonLake is certainly unusual.
The city of Taichung itself is worth staying overnight in. Pretty cool.
South Taiwan:
Kaohsiung is a vibrant and fun city , closely rivalling TAipei. But more laid back. Fun to hang out in for a few days.
Further south, spend a week in Kenting area. Taiwans premier beach area. Lots of hotels from cheap to very expensive. Spend your time in the Sun !!
EAst Taiwan= TAitung, I liked the CHIH PEN ROYAL (hotspring) hotel. But some have said its not as nice as it used to be. Dont know.
Further back up north on the East Side Hualian and its famous Taroko Gorge.
The whole east side you can do with a Train. Or take a bus from FONG KANG near Kenting up to Taitung, then Taitung up to Hualian by train and then back to Taipei by train as well.
If you have a car, driving the east coast of Taiwan is rewarding and very scenic.
THats the nutshell version.
Oh , Penghu Island on the west side is supposed to be great in summer. Iv only been there once and in winter , but I hear good things about spending a few days there.
And on the east side, near Taitung, they say taking the Barf Boat (everyone vomits) to Green Island and/or Orchid island is a trip not to miss. Or you can fly 20 passenger Dornier props there and back.
its quite a flight too apparently. Short but exciting.
Try Nanzhuang. One of the most beautiful places in the north and loaded with good food and excellent B&Bs.
We need up to date info. But i would stay a week in kenting area tho. But thats just me.