Learning to read and write the Chinese script is particularly challenging because it consists of a thousands of complex characters, and each character represents both sound and meaning.
Chinese is written entirely with Chinese characters or hànzi. To read Modern Standard Chinese you need to commit about 4,000 - 5,000 hànzi to memory.
Japanese is written with a mixture of Chinese characters or kanji plus two syllabic scripts and sometimes the Latin alphabet. Knowledge of about 2,000 kanji is sufficient to read most Japanese texts.
Korean is written mainly with an alphabetic script known as hangeul. Chinese characters or hanja are also used, but to a much lesser extent than in Japanese. A knowledge of about 2,000 hanja is essential when reading older Korean texts, which use many more hanja than modern texts do. In fact many modern texts use no hanja at all.
Most characters are built of components which represent physical things or abstract concepts. Learn what each of the components represents and try building up mental images featuring the components for each character. Include in your mental images the meanings of the characters and the pronunciation.
There are a number of books, such as Fun with Chinese Characters, which explain the structure and meaning of each character with illustrations and little stories.
When learning Japanese remember that most kanji have several different readings: usually at least one derived from Chinese and one that is the Japanese word with the same meaning. Japanese children learn the Japanese readings of the kanji first and later learn the Chinese-derived readings. This is probably a good technique to adopt, unless you know Chinese already.
Learn the correct way to write each character and if possible, take a class in Chinese or Japanese calligraphy.
Practice reading and writing characters at every opportunity.
Make some character cards with a single large character on the front and the character's pronunciation and meaning together with words and/or phrases featuring the character on the back. Carry these cards around with you and refer to them whenever you have a moment. Some Chinese language courses include character cards like this.
As you learn more characters, you will start to notice recurring themes and patterns. This will help you to guess the meanings of new characters. The more characters you learn, the easier it will become to learn new ones.
To avoid eyestrain when poring over Chinese or Japanese texts, dictionaries, etc, make sure your study area is well-lit and consider buying a large magnifying glass. I'm serious - when I start learning Chinese and Japanese I had perfect eyesight. After studying them for four years I needed glasses.
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chinEASE helps you learn Chinese characters by showing you how to break complicated them down into simpler parts.