The best time to go is go during the summer and autumn period in Taiwan, which is June to October period, where the weather is not that cold compared to other months.
As for booking of hotels, you can type the name of the hotel in the Yahoo! Taiwan and search for it. You can also book you hotel online at
http://www.eztravel.com.tw/ or
http://www.easytravel.com.tw/ (both in traditional chinese) for special deals on the above mentioned hotels. For hotel, I always stayed at Kingshi Hotel, which is located at the heart of Hsi Men Ding and its an affordable 3 star hotel and its very near to its metro station.
As for places, this have to depends on what kind of interest you have, but the few places I recommend to you are:
Martyr's Shrine, Peitou Hotspring area, Yangmingshan National Park, National Museum of Taiwan, National Palace Museum etc. Anyway, hope all the above can help you a little and if you need more help on Taipei, you can contact me by visiting my profile page.