There are many nice trails and beautiful sceneries in Kenting, such as Er-Luan-Bi and Jia-Le-Shui. There are also very nice beaches, like Bai-Sha Wan. Just go on Sundays through Thursdays and NOT during summer vacation if at all possible. It is so, so much nicer without the crowd. Lots of good value hotel options available year-round as long as you don't go during summer vacation.
I just wish the Taiwanese would respect nature more. I went to Chu-Huo scenic area and it was chained off with a sign that says "DO NOT ENTER" but everyone was inside. There are trash in the enclosed area left by people who went there to barbeque or play fireworks. Jia-Le-Shui was beautiful. I love tide pools. But can't the guides find something else to say about nature other than naming different rock formations?
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