The common language in China is Mandarin.
Think of China like the British Isles -- English being the common language, and various regional dialects/languages such as Scottish Gaelic, Welsh, Irish, and its variants. The difference being that in China the local dialects are more widely spoken than the local dialects on the British Isles.
Cantonese is just a regional dialect spoken in parts of Southern China. There's also hokkien, tiu jiu, shanghainese, hakka, etc. spoken in other provinces. In terms of the major business centers in the Chinese universe (not just China) -- Beijing (Mandarin only, like London with English), Shanghai (Mandarin and Shanghainese), HK/Guandong (Mandarin and Cantonese), Taiwan (Mandarin and Hokkien), and Singapore/Malaysia/Indonesia (Mandarin and Cantonese/Hokkien/tiujiu/hakka depending on the person).
There isn't a strict delineation anymore, although most of the time the local dialects are spoken in casual or personal situations - with family, friends, partners/spouses, or in everyday casual conversation as the situation arises. Otherwise, the default language is Mandarin - and especially in an office environment, Mandarin is used almost exclusively amongst Chinese people. Hong Kong is the exception - traditionally English in the workplace, Cantonese otherwise, but since the '97 handover Mandarin has become more widely used in addition to English and Cantonese.
As for learning the language, quite a number of the westerners have mentioned at least to me that it's not as difficult as it may seem -- at least the spoken language.
In Mandarin, the key is mastering the four tones (http://mandarin.about.com/od/pronunciation/a/tones.htm
) -- once you do that, it's actually not too hard because grammatically Mandarin is very simple.
No tenses -- "I go yesterday, I go now, I go tomorrow, I already go, I go just now, I will go..."
No verb conjugation -- "I be, you be, we be, s/he be, they be..."
Gender-neutral pronouns - there's no "she, he, it" -- just a generic "it"
No adverbs -- "it sing so beautiful" rather than "she sings beautifully"
No distinction between plural or singular -- "one apple, no apple, three apple..."
The grammar is extremely, extremely simple (i.e. the complete opposite of German). Now you know why native Chinese speakers have such a tough time with English grammar.
As for ways to learn, obviously immersion is the best (living in China), but maybe start with the Rosetta Stone series. For many westerners, the four tones is usually the hardest thing to master because there really is no equivalent in western languages. The word "go" in English means the same thing whether you say it in a high tone, medium tone, low tone, or accented tone - but in Mandarin each tone will make it a different word (the about.com link above is a good example - "ma" could mean mother, hemp, horse or to scold depending on the tone, which is why it's so important).
Reading/writing however is another beast. Most people who learn Mandarin usually separate speaking and reading/writing because they are in many ways mutually exclusive. Unlike any modern language, all the Chinese languages don't use phonetic alphabet - it's all characters (like ancient Egyptians with their hieroglyphs). To learn how to read/write Chinese characters, you'll learn it much the same way Chinese kids learn it - straight up memorization. There's no patterns or anything - it's just memorizing. To be able to read basic everyday stuff (street signs, menus, etc), you'll need to know at least 1,000 characters (each character is a word, and multiple characters can also be a word but there is no way to know other than context -- if let's say each letter was a "character" or "word", then in Chinese a sentence would be written this way: "I a m g o i n g t o t h e s t o r e"). To be able to read a newspaper comfortably or business correspondence, you'll need around 2,000 characters. Most native Chinese speakers know around 3,500-5,000 characters -- writers/novelists/literary minded folks may know up to 10,000 characters.