Where is the best place for an European student to live and learn Chinese?

[Brat]

You can go to China and rent a VPN to tunnel through the Great Firewall of China.
Or you can go to Taiwan.
Do NOT go to Singapore as suggested, the official language there is English and a large part of the population does not speak Mandarin (Less than half speak Mandarin as a first language and almost everyone is fluent in English or at least Singlish, the local variety, that means you will probably not learn any Mandarin and screw up your English at the same time) - they have internet censorship and a brutal society, although you will find lots of nice and friendly people there. And you really need to obey the law there - death for illegal drugs and several years of prison for things considered minor offences in Taiwan and China. Rent will be far higher as well.

In China you will be able to learn Standard Mandarin as spoken by the majority of Chinese speakers in the world - the Han Chinese in Mainland China. And you will learn simplified Chinese Characters, the form of Chinese characters you will find most texts written in (most Chinese abroad will use traditional Characters though since the reform was not long ago). It also appears to be easier to read traditional Chinese for those who know simplified, rather than the other way round - although one would think the opposite. Simplified is, as the name suggests, a whole lot easier than traditional.

Mandarin is quite different in Taiwan, although the dialects are mutually intelligible. It is true that Taiwan has preserved more of the Chinese culture than Mainland China has, including traditional characters. The people are generally more open, more relaxed and more educated than in Mainland China, but you will find amazing hospitality and also good friends in both countries. Taiwan does no Internet censoring that I have heard of, but if it is a limited period of time you are spending in that place, should this really have an impact on your decision? Don't you want to spend your time learning a new language, rather than on Facebook and Twitter?

Europeans are accepted and respected and treated very friendly and hospitable in all three places, but you will make experiences of being stared at for looking different mostly in Mainland China. That is not necessarily a bad thing because it often opens doors for you - especially Mainland Chinese are often overly hospitable towards foreigners - but you need to get used to it.

If Internet censorship is an absolute no-go for you, choose Taiwan. If you want to learn the more useful version of Mandarin and Chinese Characters, go to China. But I repeat, learning Chinese in Singapore might work but it is a BAD idea.