恭喜!恭喜!gōngxǐ gōngxǐ '(respectfully) wish (you) happiness'
恭喜发财!gōngxǐ fācái 'Wish happiness and prosperity'
新年好!xīnnián hǎo 'New Year Good'
新年快乐!xīnnián kuàile ' New Year Happy'
What can you wish people?祝你。。。zhūnǐ (wish you)... +
身体健康 shēntǐ jiànkāng 'Good health'
万事如意 wànshì rúyì 'Everything as you wish'
Married People or Senior people will give 红包 hóngbāo (Red Envelope containing cash) as a symbol of luck to unmarried people, kids or lower-ranking people (like employers to employees).
Here is a popular song called Gongxi Gongxi with Pinyin that you will hear in every Chinese community during Chinese New Year:
Learn more:
http://laowaichinese.net/text-message-greetings-for-chinese-new-year.htm
http://mandarin.about.com/od/chineseculture/a/newyear.htm