There are many cell phone service providers in Taichung.  The rates are often very similar unless you have several friends using the same service.  Then it makes sense to get the same service provider that they have as you will get discounted rates for calling on the same network.  Unlike America, the cost of the call is billed to the caller only.  It does not cost you anything to recieve a phone call.  Taiwan Mobile and Chunghua Telecom both offer English services. 

If you have a friend who has an old cell phone around (almost everyone does) you can use their phone and get a what is called an "IF" card from 7-11 or any other convenience store.  The difference between this and getting your own service is that it is more temporary.  All the cards are prepaid and therefore you dont have to enter into a two year contract which is normally required.

If you know you will be staying in Taiwan for at least two years, it makes sense to apply for your own number.  The calling charges are significantly cheaper and you can get a new phone for quite a bit cheaper if you buy it with a plan.

What you will need to apply:
1.  Your passport
2.  Your ARC
3.  In some cases, a cosigner (depending on which company you apply with)