You could call around to some travel agents of course (many also place ads in the English papers available at 7-11 and other places...) or stop in at some agency office you stumble upon in the city. There are indeed a number... Asking locals may be useful to direct you to more economic agency. You could try www.zuji.com.tw as well but will need help from local as in Chinese. There are some other Chinese sites too, but can't think of English ones. You might try www.evaair.com to buy direct from that airline. They are often as reasonable or more so than agents anyway. China Airlines, the state airline which is the other big one, has a somewhat disturbing safety record. I am afraid there aren't any true discount airlines flying to Taiwan at this time. There are also smaller airlines with some limited routes and English sites:
http://www.fat.com.tw/en/Ewebpage_beta3.asp
http://www.mandarin-airlines.com/en/exfile_index.htm
These may or may not be good to buy directly from, but they are good to see some schedules on, and subsequently remind
agent to check pricing of too... Of course, there are plenty of foriegn airlines to travel from Taiwan to other parts of Asia, but this would be considerable trouble to check out.
Would be good to get a local to help you look at aforementioned site or yahoo.com.tw. Otherwise, or in addition, call around to the agents in the paper (remembering to ask if price includes all taxes...) and maybe drop in on an agent you see or someone recommends. Another interesting thing about a lot of the agents you might stumble on in Taipei though is some will have their own sites and preferable rates if tickets bought online. Take card and get friend to help you check if no English.