Taiwan and Korea rival one another in Asia with respect to the amount of time spent on-line. Almost all new apartments in Taiwan come hardwired with broadband making keeping in touch with friends and family at home, planning trips, researching culture and lesson planning super fast and ultra convenient. In Taiwan we haven't heard of dial up!
The average cost of a broadband connection with unlimited access time in Taiwan is NT$500/month.
If you don't get hooked up at your apartment (a rarity these days), you will always have internet at your school and internet cafes are ubiquitous on the streets of Taiwan. Time is dirt cheap, most charge only NT$15-30/hour (USD 50 cents- $1).
Should you need to set up internet on your own you will need your passport and an ARC. Even with an ARC you may still need a Taiwanese guarantor - too many foreigners have left without paying.
Many landlords will set the internet up for you and will charge the monthly fee as part of your rent. This makes things easier for everyone involved (and is very nice of your landlord!).