Mandarin Chinese refers to a group of languages traditionally found in northern China. Mandarin languages can vary as much as French, Spanish and Italian, but the word 'Mandarin' typically refers to the official language of China, based on the Mandarin language found in Beijing. Taiwanese is just standard Mandarin with a Taiwanese accent. They're considered the same language.

Shanghainese and Cantonese are not only separate languages, they're in separate language groups, like French, German and Russian. Shanghainese is traditionally spoken in Shanghainese,  but nowadays many locals from Shanghai can't even speak it, only knowing standard Mandarin.

Cantonese refers to the traditional language of Guangdong (formally known as 'Canton'). People in Guangzhou generally speak Cantonese informally with each other, but they can all speak standard Mandarin since they grew up hearing it on TV and in school. People in Hong Kong also speak Cantonese,  but they don't necessarily speak Mandarin. I've heard that more people speak English than Mandarin in Hong Kong. There are also political tensions in Hong Kong, where people might get offended and get angry at you if you speak to them in Mandarin. They'd rather speak to you in English or Cantonese (this applies to people from mainland China as well - they speak with them in English rather than Mandarin). Unless you're speaking to people from Hong Kong, you won't need to learn Cantonese.

In the past, Cantonese was the official language of China. Apparently some ancient poems rhyme in Cantonese but not in Mandarin

It's worth noting that Chinese people often call all these languages dialects, even though they're distinct languages. This is in part because the Chinese word '方言' doesn't quite correspond to the word 'dialect' (English and Hindi could be considered 方言 in India, even though they're very distantly related), and partly because they use the same writing system (because characters are not based on phonetics). Well, they used the same writing sytem until 1956, when the PRC officially introduced a new set of simplified characters, replacing the old ones. For example, 個 became 个. Traditional characters are still used in Taiwan and Singapore, since they've never been part of the PRC. Traditional characters are also used in Hong Kong, which only became part of the PRC in 1997. So the Mandarin and Cantonese in Beijing and Guangzhou respectively both use simplified characters, but the Mandarin and Cantonese in Taiwan and Hong Kong respectively use traditional characters.

Thanks for your complete explanation,Jmat.