Is Taiwan language same as Mandarin?

I love China and Chinese culture. I watch Chinese dramas with English subtitles on TV every day.
Since I want to understand what they’re talking in Chinese directly, I am interested in learning Chinese here in the U.S. But I wonder if Taiwan language is same as Mandarin or not. Can I use Mandarin in Taiwan? Do they understand?
If I start learning either of them, which one do you recommend?
Dear eido 1234, glosandro,Durian, and drshorty:
Thank you so much for giving me such wonderful answers with details!
How can I choose only one person for "the best answer"? Every answer is the best to me.

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