Let us start with the commonly used greeting which is "Happy New Year"

Literally translated this means 'Congratulations get rich'. Of course metaphorically, it means 'Wishing you prosperity' and is used as a CNY greeting.
There are many words and greetings which on the surface appear to say one thing whilst having a deeper hidden meaning. Let us look at another example:

When literally translated, this saying means 'ten thousand things according to your will'. Metaphorically, it means 'May your wishes come true'. The character 'maan' is the word for ten thousand, which is a Chinese unit. 'See' is the word for things or happenings. 'Yuu' means 'according to' and 'yi' can be put together with other characters to mean different things but in this case it's understood to mean 'what you want'.
And one of the most common characters you will see around and about during Chinese New Year - Fook

Fook can literally be translated as 'Blessed'. But it does have deeper significance which can only be expressed in English through combining several words, such as 'good luck', 'being fortunate', and 'auspiciousness'. That's why you often see this single character used on many things, especially during Chinese New Year, where this character is stuck on doors, printed on red packets (lei see envelopes) and many other things.
Here are some more with the literal translations attached:
龍馬精神
lung ma jing sung
(meaning Spirit of dragon and horse)
心想事成
Sam seung sih sihng
(meaning accomplish that in your heart)
恭賀新禧
gong ho san hei
Congratulations on new blessings
大吉大利
daai gat daai lei
Much luck and much prosperity
五福臨門
ng fuk lam mun
five happinesses bestowed on your household
出入平安
chut yap ping on
leave and enter in peace and safety
年年有餘
nin nin yau yue
every year have bounty in excess
for children:
快高長大
faai gou jeung dai
quickly grow taller and become bigger
添丁發財
tim dang fat choi
add more (male) heirs, and become prosperous
招財進寶
jiu choi jun bou
seek wealth welcome in the precious
生意興隆
sang yi hing lung
let the business be popular and prosperous
盤滿砵滿
pun mun but mun
basins full and bowls full (of wealth)
笑口常開
siu hau seong hoi
laughing mouths opening frequently
天天向上
tin tin heung seung
daily heading upwards!
Wishing you prosperity, may all your wishes come true and may you be truly blessed.
Happy New Year!!