This was one of the hardest things to do in Taipei, while much easier in the south of Taiwan, due to simple supply and demand economics. When I lived in Taichung and Kaohsiung, renting an apartment was as easy as calling the phone number on a window sign next to the character for "for rent" or “租”.
Arrange a viewing, preferably with a local taiwanese friend, make sure you like the place, put down two months deposite and one month's rent, and grab the keys.
Taipei is another beast. With so many people looking for an apartment, and so few nice places available, you need to be resourceful. First, bring a Taiwanese friend who looks the part of a hard-working, no nonsense kind of guy or girl.
When you meet the landlord at the viewing, mention that your life is consumed by work (or study) and you really have no time to party of socialize with friends. Landlords are worried that expats will have loud parties all the time and disturb the neighbors (and rightfully so). So it's crucial to give off an image of not being this type.
Secondly, don't take even a minute to consider if you'll take the spot, because if you hesitate, someone else will swoop in and rent it. Bring a wad of cash to the apartment viewing and if you like the place, put the deposite down on the coffee table and show the landlord you mean business. Work on signing a contract right at the viewing and grab the apartment of your dreams.
By the way, the website 591.com.tw is all you'll need for setting up viewings.