What do you speak, TheElements? @AuldAlliance
Well... 
My family are from Penang (one of the original Crown Colonies) and of Nyonya descent. Like most Nyonya families, there has been plenty of introduction of "fresh" blood of more recent Chinese migrants.
So on my dad's side, I learned Teochew (Chaozhou) dialect whilst on my mum's side, I learned Hakka dialect. Because we are from Penang which is in the north of the Malay Peninsula, we also speak Penang Hokkien dialect. Plus, since this was Malaya after all, everybody also spoke Malay. All this was passed down, informally, as it had been for generations as a collection of pidgins. In fact, our preferred language amongst family members and other Penang Nyonyas is English albeit a distinctly Malaysian version.
My family were also formally educated in English which was the language of the colonial administration and a lingua franca for all the ethnic groups in the federation, which persisted even after independence. Also after independence, Malay was declared the official primary language and subsequently everybody was also educated in Malay.
My parents moved us to the capital Kuala Lumpur for work when I was tiny. KL is in the "middle" of the Peninsula, where the main Chinese dialect is Cantonese. So I had to learn that. Although Mandarin classes were offered in school, they were rather patchy in quality and my parents weren't that bothered so didn't push us to learn. And then I moved further south to Singapore in my teens to study, where the main social dialect has been Mandarin since the (?) early 80s (?) so I had to learn then. BTW, the elderly Singaporeans would often still speak Southern Hokkien dialect too.
All this has changed a lot since the rise and rise of China as a superpower, and SE Asian ethnic Chinese people just a few years younger than me tend to have embraced Mandarin education, eschewing their ancestral dialects except for chatting to their aged relatives.
So: I speak several fossilised Chinese dialects, to varying degrees of fluency and pidgin-ness, but am hardly able to read the written characters. Whereas thanks to formal education (and actual proper daily use) I'm first-language fluent in English and Malay.
Upshot: Many ethnic Chinese people take the trouble to clearly distinguish between which dialects they mean, because it really matters. We don't really ask each other "Do you speak Chinese?" but rather "Do you speak X dialect?" - I wouldn't say the dialects are completely unintelligible as you can generally see the relationships between the words (and the grammatical structure is basically the same), but you'd have to both work really hard, speaking slooooowly and listening carefully, to get the meaning across.
Further tidbit: Although Taiwan and the PRC both officially use Mandarin, there are some variations and they use different versions of Chinese script. Taiwan still uses traditional characters whilst the PRC use simplified characters.