Since Taiwan is a subtropical country in full development, health conditions are quite different from the ones you were used to. The summers are hot and humid and with the introduction of air-conditioners many people catch summer colds. The winters on the other hand can be chillier than one would expect and since there is no heating provided, you should learn to mix and match several layers of clothing to stay in good health.
As a rule of thumb, it is good to remember that tap water in Taiwan is not very safe to drink. The water fountains (such as the ones on campus) are regularly inspected, but otherwise water should always be filtered and/or boiled before consumption.
To prepare medically for your stay here, it is advised to have the following inoculations and/or vaccinations: tetanus, poliomyelitis, Hepatitis B and Japanese B Encephalitis. Check with your local doctor for more up to date information. If you really want to be prepared well, you might start getting these injections back home, because some of them don't come in one shot only, but take a series of three or four shots with a few weeks interval. If not, all these vaccines can be obtained safely and efficiently here in Taichung. You might start taking the shots at home and bring the remaining medicine over to have them administered here.
Every hospital has extensive outpatient services. Registering and finding the doctor you want to see (each doctor works three half days a week), as well as paying fees and getting medicine is a total zoo, so you must go with someone who understands the system. Expect to find competent, even accomplished doctors (based on your friend's recommendations) giving treatment very quickly in assembly line fashion.
Jen-Ai (pronouced Ren-Ai) is the nearest to NCHU and strives to be the most foreigner friendly in the area.
Clinics are all privately owned. There is no appointment needed, though some clinics do have them and you waste less waiting time there. The doctor doesn't necessarily charge for treatment per se, but for the medications he or she dispenses (usually 2-3 days' worth). You are expected to come back for more medications after yours run out. You may ask the names and strengths of the pills you are taking, but the doctor may be reluctant to give them to you for fear you'll go to a drug store and buy them yourself. Though since a few years ago doctors officially don't have the right anymore to dispense medicine, most clinics still dispense the medicine themselves.
As it stands right now, you don't need a prescription to get most drugs here. With the help of the staff at the Language Center or of other Chinese friends, you can easily enough find licensed pharmacists around the University. For minor illnesses the pharmacists are quite happy to give you advice and prescriptions, but be aware that they don't usually ask what other medications you may be on (which could cause synergistic effects).