I stayed in Taipei for 4 years, and I really loved it there.
In Taiwan they speak Mandarin, people often refer to their accent as "Taiwanese", but that is false, Taiwanese is a different dialect of Chinese that is only really spoken in secluded places by grandparents. Everyone speaks Mandarin.
I found Taiwan to be more modern in terms of their infrastructure than suprisingly enough, Seoul... So that is saying something.
The people are nice, as long as you actually make an effort to integrate into their culture and learn proper Mandarin.
While it is significantly more westernized than the rest of China, English still isn't spoken all that widely, but that is changing very quickly as English is now a mandatory course in their school curriculum now.
BUT, if you are sensitive to being made fun of, or "bullied", then don't go there; they won't hesitate to call you a 老外 (lǎowài). A good amount of the people are fairly xenophobic, but it isn't much of a problem, if you learn Mandarin they will accept you with open arms.
If you have any questions feel free to ask.
It is under direct control of the PRC
So yes, technically it is part of China.
BUT, it retains de-facto independence in some issues.