While the obivous place to go to learn Chinese is, well, China, it's sometimes easy to forget that for quite a while after the communist takeover on the mainland, Taiwan (officially the Republic of China) was the only "China" recognized by the Western world. Global politics aside, Chairman Mao's victory has resulted in some lingering after-effects for Mandarin Chinese learners as well.
First of all, the flight of China's imperial government to Taiwan in 1949 introduced Mandarin as the principal means of communication across the island. The ensuing complete political separation between Taiwan and mainland China meant that Taiwan's writing system never underwent the same simplification the communists enacted to increase literacy on the mainland. Now, only Taiwan and Hong Kong continue to use traditional Chinese characters, while China itself has fully switched to simplified characters.
Due to the United States' history of good relations with Taiwan, coupled with the mutually held suspicion of China during the Cold War, Chinese language instruction in American schools and colleges continues to emphasize learning traditional over simplified characters. American students learning Chinese might therefore find studying in Taiwan meshes more easily with their program of study. Even for those contemplating studying abroad without prior language experience, Taiwan is an excellent option to consider. It is much easier to learn simplified characters after having studied traditional, but more difficult to reverse the order.
Taiwan has more attractions than simply its writing system to offer the curious student. Its capital of Taipei is a bustling, cosmopolitan city with pan-Asian cultural influences. Past colonization and rule by China, the Netherlands and Japan coupled largely autonomous rule after Word War II have resulted in a rich and unique culture and a certain local pride separate from mainland China. Taiwan also was not subject to the cultural devastation wrecked by Chairman Mao in the early decades of Chinese Communist Party rule, so many historical artifacts and some practices like traditional puppet theatre were preserved in Taiwan. Taiwan's government has instituted democratic and liberalizing reforms over the past few decades, so the difficulties of internet censorship, restrictions on free speech, and political sensitivity on the mainland do not apply to Taiwan.
The two major cities in Taiwan are Taipei and Tainan, although several good language institutes are located in Taichung as well, and the countryside of Taiwan is notoriously beautiful.