Taiwan is beautiful throughout actually. Each area has its own beauty.
In the north you have Taipei. It is a city much like KL, but there are many things to see. You have the beach in Baishawan beach and Fuji harbour (great seafood). You also have Danshuei and Bali. The fisherman wharf here is just awesome. The city itself is interesting with the many streets and night markets, the biggest being the Shilin night market. There are others, smaller one, but with different things to see.
You also have specific street catering to specific needs, like camera street, watch street, books street etc. You also have Xinmending. It is an area for young people with many interesting things to see.
An hour from Taipei you have Ilan. You will have to pass through the 13 km long tunnel to get there. Ilan and Suao (where the cold water spring is a constant 22 degrees, regardless winter or summer) are very beautiful. It is very flat and very different from Taipei and anywhere else in Taiwan.
Further east you have the Toroko Gorge. It is a must see place. Awesome. You also have Taidong and Hualien. You are into driving, the Suao Hualien drive is a drive you must do. It is long and dangerous, but the view is unbelievable.
In the west you have Hsinchu. The wind city. If you like kite flying this is a place to go. It is a beautiful place as well.
The next city is TaiChung. One of the places to visit is Gaomei wetlands. If you have kids, a place to visit. Lots of hermit crabs and fiddler crabs. Taichung is a new city, so the road are wider as compared to Taipei. Better planned as well (some say).
Moving south the next city is Tainan. The food here is better as compared to Taipei (some people say, personally, it taste about the same).
The next city is Kaohsiung. This is like the cowboy city in Taiwan. The people here are more friendly, but rougher. The people here speak more Taiwanese (hokkien). Driving here is an experience as many people here consider traffic lights a mere suggestion.
The last city is Kenting. The beaches here is just awesome, as is the aquarium. Summer is the best and worst time to visit Kenting. It is also the place where they shot the movie Cape No. 7.
In between you have Sun moon lake, Ali Shan Mountains, etc....