Your goals for your time in Taiwan should help you determine where you study Mandarin in Taiwan. The country offers plenty of public and private schools that offer very different learning environments. I've spent time at three different schools studying Chinese in Taiwan, and have heard plenty of casual reviews from other programs.If you are set on Taipei for study, the two schools that most people look at first are the language centers at National Taiwan University (NTU) and National Taiwan Normal University (NTNU), which are basically just down the street from one another.
NTNU has a reputation of a slightly more relaxed environment, while NTU has a reputation for a somewhat more demanding program. Both are relatively affordable, and will give you a good working knowledge of Mandarin within a year, if you can commit to two semesters of serious study. The same goes for most universities with Chinese language centers. If you are in a rush, and need to get your Chinese at a working level as soon as possible, then NTU has an accelerated program that will bust your balls for about eight months I think. I've heard people say that program covers about the same amount of content as a year and a half worth of study in the regular programs.
But I don't won't to oversell schools in central Taipei when there are dozens of great options to choose from across the island. Outside Taipei, I can vouch for the effectiveness of programs at Tunghai University and Fengjia University in Taichung. Tunghai has small class sizes in a very green cozy campus in central Taichung and is a solid choice for the casual Chinese learner that has reservations about living in Taipei. Fengjia is in more of a central urban environment and is a good place to find an active social life as you pursue your studies. Another good choice outside Taipei that comes to mind is National Sun Yat Sen University (NSYSU) in Kaohsiung. That school is a favorite among a lot of foreigners, because it is literally next to the beach (students can easily hit the beach before and after classes there).
Speaking from personal experience, if you are starting from scratch and your goal is reach genuine communicative competency, I'd recommend two years of regular study at a university. But I understand, that is a big commitment and unrealistic for a lot of people visiting of living in Taiwan.
Other options available are the private run language academies that can be found in pretty much all the large cities. The Taipei Language Institute (TLI), Taipei Mandarin Center (TCM), and Taiwan Mandarin Institute (TMI), are the ones that come to mind. The language academies offer a somewhat cheaper option than the university programs, but they don't offer student visas for people studying there. These are a good option if you are just coming to Taiwan on a tourist visa and want to give studying the language a go for a few weeks or something. These schools can easily accommodate irregular scheduling and by offering one on one classes or small group classes, they can tailor the class curriculum to the students' needs and levels.
However, the quality of classes at the language academies may not be as consistent as the classes at the universities. It really just depends on the teacher you end up with. And that's not to say there aren't good teachers at the schools. They just draw from a much wider pool of instructors than the teachers at the university language centers The universities tend to employ more career-minded educators while the language academies employ more part-timers, or recent graduates.Anyways, those are a few things to think about if you're headed this way or already here looking for a place to study.