The rules depend on which country your passport is from. I'm American, and came here on a 3-month HES scholarship last fall. They gave me a "visitor visa for purposes of studying" (note: there is no "student visa"). For Americans, it's a 60 day visa, renewable twice (total of 6 months). This means that every 60 days you go to the immigration agency and renew it. DO NOT forget to do this. I was late by one day and miraculously had no consequences, but I've never heard of anyone else being so lucky. Most people I believe have to pay a fine and do an expensive visa run, and I've even heard of people being deported. Anyway, MTC has all the paperwork you need for that in the 6th floor office. After you've been in class (not in the country, but actually attending class) for 4 months (I think), you can apply for a resident visa. A resident visa is good for 15 days, by which time you need to apply for an ARC (alien resident card). This is an involved process (including a medical exam), so get started on it early so you don't overstay your 6 months. This is all done within Taiwan, so it isn't true that you can't change your visa in country.
Of course, don't take my word on all this. MTC (and any other school, I'm sure) should have all the info you need, and should be more authoritative.