Taiwan is a land of many fascinating landmarks. Arguably the best is the National Palace Museum. Schedule a full day to enjoy the vast combination of art and history.
JuMing is a world famous artist and sculptor in Taiwan that has an entire museum dedicated to his works. Many are outdoors and some are literally as big as a small house.
In Taipei there is the equivalent of the "tomb of the unknown soldier". The changing of the guard happens each hour. During the hour the guards stand perfectly still without even the slightest facial expression.
A colorful monument in the beautiful southern city of KaoHsuing. In the south there is Kenting park. The southeast of Taiwan has some of the most beautiful coastline in the world. The ocean is a pretty bright blue and the coastline is a rugged mountainous terrain.
Two of the best known landmarks around Taipei are the Grand Hotel and the 101 skyscraper. The 101 building is so tall that you can usually use it as a reference. Taipei is surrounded by mountains and ocean and they say it sort of sits in a bowl. The Grand Hotel is a beautiful building with red columns and gold roof. The hotel has been a historic landmark for many years.
You should try to stay at the Grand Hotel at least one night. However, if you are planning a trip and if at all possible schedule some time to stay with friends or families in their homes to get a feel for the real Taiwanese lifestyle. Taiwan is a beautiful island with very friendly people. You must understand that life there will be very different from what you are accustomed to if you live anywhere in the United States. It may be best to consider a trip there as a challenge as opposed to some time off to relax.
Speaking of challenges, Alishan mountain is a tourist favorite but the longer treks were strenuous with a lot of steps going up, up, up!