What are best spots to see and things to do near Taipei ??
That's rather too long a topic for one thread. You can get a good idea by reading through the posts in this forum.
You should also get a guidebook. The Lonely Planet and Bradt guides are both good; and their authors frequent this site.
Oh, and not all of Taiwan is beautiful -- especially the west coast and the cities. But plenty of it is.
And, yes, do go to Tainan. For someone interested in culture, a trip to Taiwan without seeing Tainan would be incomplete.