If you are looking for a language center to study Mandarin, I would recommend the Mandarin Training Center of National Taiwan Normal University, they provide best language courses and teachers in Taiwan. Many foreigners go to Taiwan and take Mandarin courses at this center. If you are looking for a degree program, National Taiwan University should be your first choice for they have the most resources and also the top ranked university in Taiwan. There is also a language center in the university. Both universities are in Taipei and are not far from each other. In fact, I believe it takes less then 10 min to take bus from one to the other. For more detailed information, just google and go to their websites. Just to mention, if you have learned Mandarin in simplified Chinese, it is not taught in Taiwan. We read and write traditional Chinese in Taiwan.

"很高兴认识你" would be the direct translation and also a quite common usage in Chinese. Notice this is only used when meet someone at the first time. If you are talking to someone you already know, (e.g. Nice to meet you again, Ben!) you would probably say "很高兴见到你" or "幸会" (the latter is less used and a very formal expression)

Most foreign visitors go to 故宮(National Palace Museum), 中正紀念堂(Chiang Kai-shek Memorial Hall) and 台北101(Taipei 101). It is also recommended to go to 士林夜市(Shilin night market) where you can have all kinds of snacks. If you prefer being close to Nature, go to 陽明山 (Yangmingshan National Park). You may also visit 龍山寺 or 行天宮(most famous temples in Taipei) if you are interested in traditional religious buildings. Citizens in Taipei also like to visit places in more suburb areas like 九份(a fascinating mountain village, must try some delicious 芋圓 if you go) and 淡水(a riverside town). If you are going shopping in Taipei, of course there are many shops, but those shops are quite similar as you may see in other big cities in the world. Therefore I would recommend you to have a food guide in hand and visit some local restaurants. Wish you a pleasant trip in Taipei!