There are lots of options for learning Chinese (Mandarin) in Taiwan. In general, these could be separated into the free options and those that cost money.
Free Options:
1) The IAA offers free beginner classes on Monday nights (and advanced classes on Thursday nights), taking place from 18:30–20:00. There is a refundable deposit of ~ 1,500 NTD for the course but if you miss more than a handful of lessons (even for reasons of business travel) you won't get the money back and may not be allowed to take further language courses. The lessons are held at the TIGP building near the Academia Sinica campus out at Nangang, which takes about an hour to get to from NTU campus by MRT. These lessons are useful for providing a basic grounding in Mandarin and a minimal vocabulary. However, many people quickly find these classes to be of limited use and move on to other alternatives.
2) Language exchange. Lots of Taiwanese are looking to learn English (or Japanese, Spanish, German etc.) so it is relatively easy to find someone interested in sitting down with you to help you with your Chinese in exchange for your help with their foreign language skills. Some good places to look are the Facebook language exchange group, and tealit.com. Be advised that a substantial fraction of the language exchange activity on these sites is more like online dating, but there are also a lot of people out there who are just trying to improve their language skills.
3) NTU offers Chinese classes for foreign exchange students and some of us have infiltrated these classes in the past as non-registered students. The way to do this is to contact the Chinese Language Department at NTU cld222 [at] ntu.edu.tw to ask about scheduling of classes. Once they have a schedule they can tell you where and when the classes are to be held. You can then just show up on the first day and ask the teacher if you may sit in on the class. The night classes are usually twice a week from 18:30–21:30. Daytime options also exist, but may conflict with your work schedule.
4) By yourself! Lots of good online material exists... check out the following links (just a few examples):
5) Useful phone/tablet apps:
Pleco Chinese/English dictionary (Android+iOS)
HelloChinese language learning app (Android+iOS)
Memrise language learning app (Android+iOS)
Duolingo language learning app (Android+iOS)
Paid Options:
Many options exist to pay for Chinese classes in Taipei. The major universities offer all levels of Chinese. Check out this listing of university language centers. If university classes don't fit your schedule or your budget, you can find a private tutor for between 300–700 NTD per hour on the same Facebook and tealit pages mentioned above. Private or group classes can be arranged at any of the private language institutes in Taipei for a similar price.