Don't use your chopsticks to point at someone, or gesture wildly about with chopsticks when you're talking.
Never stand your chopsticks upright (such as in a bowl of rice) because it is a custom associated with offerings to the dead.
Never use the pointed end of your chopsticks to pick your teeth, or to scratch yourself.
(There are more etiquette rules presented on http://www.thingsasian.com/goto_article/article.2418.html, but I found this
last one [Don't scratch yourself, Homer] particularly funny).
One interesting observation I've made is that in Chinese restaurants in North America, often the ethnic Chinese will use forks when eating off flat plates, while Caucasian companions or groups will use (or attempt to use) chopsticks. Chopsticks are actually not that practical when using plates, but are more useful for bowls (to scoop with). NOTE: The more glutinous Japanese rice is much easier to pick up than the long-grained Indian rice, and would be better to practice with.
The official national language of China is Mandarin (Putonghua = common speech), which refers to a group of dialects spoken across Northern China. The Beijing dialect is considered the standard (such as Hoch Deutsch is for German), but the common spoken Mandarin has a much softer sound without all that tongue twisting that the strict Beijing Dialect contains. The Mandarin Language is referred to as Guoyu, Hanyu or Huayu, where "yu" sounds more like "yuee" with the lips rounded.
This brings up the issue of spelling of Chinese words. Why did "Peking" become "Beijing", and "Tsingtao" beome "Qingdao" and "Hsi-An" become "Xi'an"? The Wade-Giles system (Peking, Chunking, Hsi-an) has been around since the late 1800's and was a romanization of Mandarin Chinese words as they sounded to linguists at that time. In the 1950's, the mainland Chinese developed a new romanization system called "pinyin" (note that this is romanization not anglicization such that many words to an English reader will not look like the sounds at all - e.g. Xian [the city of the First Emperor with the thousands of Terra Cotta soldiers in his Tomb], which is pronounced something like "Shee Un" in English rather than "Ex Ian"). The pinyin system has been accepted internationally, but the diehards in Taiwan still stick to the Wade-Giles system (I blame them for hundreds of students constantly mispronouncing my last name [Hsiang pronounced shung4], but the pinyin spelling [Xiang] wouldn't be much easier to figure out either).
Mandarin Chinese has four tones 1=flat, 2=rising, 3=rounded (down and up), 4=falling, and when you see words with numbers written after them e.g. Bei3 Jing1, these usually refer to the tones. Foreign speakers of Chinese often fall into a single tone (such as the first flat tone) to say all the words. I guess it's hard enough to remember the sound, not to mention the tone associated with that sound.
After the revolution (or the liberation as the PRC likes to call it), Mandarin was made the official national language. Although the written language is (was) the same for all the Chinese dialects (and can also be read for the most part by the Japanese and classically trained Koreans who adopted the written words centuries ago), the Chinese dialects are not necessarily mutually intelligible, and can differ as much as English and German do, although some are more similar, such as the difference between Spanish and Italian. Under the communist regime, there was an attempt to simplify Chinese writing (reduce the number of strokes in complex characters). These simplified Chinese words are used by mainland Chinese and some other groups such as the Singaporeans, but Taiwan and Hong Kong still retain the traditional Chinese writing
The national language of Taiwan is also Mandarin, but the standard accent is different. In Taiwan, the dialect of the local Taiwanese people, which is a variant of Fukianese (Min Nan Hua subdialect) spoken in the Chinese province (Fujian) closest to Taiwan (and the major source of Chinese migrants to Taiwan historically), is becoming more dominant even in official circles with the rise of the Independence Parties in Taiwan.