For Chinese speaking people like me, generally we don't say 'speaking Chinese'. The term of 'Chinese' is too generic. It is 'Speaking Mandarin', 'Speaking Shanghainese' or 'speaking Cantonese', to be more precise.
Standard Chinese Language
The official standard Chinese language is Mandarin or Huáyǔ 华语, as we Malaysian like to call.
China refers the language as Pǔtōnghuà普通话. It literally means 'common language'.
The Taiwanese Chinese refers it as Guóyǔ 国语 which is 'national language'.
Chinese Languages/Dialects
Other than Mandarin, Many Malaysian Chinese speak Hokkien and Cantonese.
Hokkien* is the dialect spoken in my family, considered my first language. I learnt speaking Mandarin in Chinese primary school.
Huge transformation happened in secondary school (after age 12) where school system had our national language (Malay/Bahasa) as the main language. Everything converted to Malay. Exams papers were all in Malay. Learning Chinese language was solely on our own at this stage.
When I moved to the capital Kuala Lumpur, I had to speak Cantonese to get by. Learning it was not difficult due to the popularity of Hong Kong drama series and pop songs.
In Malaysia, some English educated Malaysian Chinese don't speak Mandarin. Most of them are fluent in Cantonese simply because they grown up speaking Cantonese. Compared to them, I still have problem on mastering its six phonetic tones!
A multilingual Malaysian
All these make Malaysian Chinese able to speak more Chinese languages than the rest of Chinese origin people in the world. Not that I specifically proud of this but it is just the way to survive in my country.
How about you telling me your experience of learning languages?
Read the trouble with my Chinese name and what makes a Chinese name if you are interested in understanding a Chinese name.
*'Hokkien' in Chinese is 福建, a province in Southern China which named 'Fujian'. Malaysian Chinese and Singaporean Chinese refer the word 'Hokkien' as a language/dialect but not for the Chinese in China. To them, Mǐnnányǔ 闽南语 is the dialect, not Hokkien.