What do they speak in Taiwan?

They speak both Mandarin Chinese and 70% of the population also speaks Taiwanese Hokkien. Both are official languages of Taiwan.

Is Taiwanese the same as Mandarin Chinese?

No. Taiwanese is a different language ( Taiwanese Hokkien ).

Are the characters the same in Taiwan and Mainland China?

The same characters are used but the difference is that in Taiwan traditional characters are used, whilsy in Mainland China simplified characters are used. There may also be some pronunciation differences between the two.

Is the Taiwanese Mandarin accent different?

Yes, the accent is often referred to as much softer in Taiwanese mandarin and the 'shi' and 'si' sounds can sometimes sound indistinguishable from one another!