Taiwan is still pretty big as a category, especially if you're looking at south of taipei. Taiching(food), Kaosiung(monuments&food), Hualien&Taitung(scenery and culture), or even Alishan. It really depends on if you're after food, monuments and picture taking, or culture, or shopping. But do watch out for crowded train station near new year as MANY people will be heading south and might cause a cram at the station, but your tickets should be fine.
If you are truly traveling solo, consider the HSR or TRA tourist pass to help you get around easily outside of Taipei as it time consuming to move without a cab in Hualien and etc.
