It's really common nowadays for English speakers to use serious abbreviations when texting each other. Even the current anti-depression campaign is called R U OK?
In hacker parlance, words are designed to confuse people, like the ones used by the 1337 (i.e. the "leet" or "elite" in hacker parlance).
So you might wonder how this plays out in Chinese, with their much more complex character sets.
Turns out that there are some really interesting variations on this.
520 is a common abbreviation for "I love you". The phrase "I love you" in Chinese is 我爱你。(Wǒ ài nǐ.)
Now getting to 520 from Wǒ ài nǐ is quite a stretch, yet it's become commonplace. 520 is 五二零 (Wǔ'èr líng) when written character by character. (The value 520 is actually 五百二十 or Wǔbǎi èrshí meaning 5 hundreds and 2 tens).
The idea is that Wǔ'èr líng sounds a bit like Wǒ ài nǐ. As I said, that's actually quite a stretch.
Here are a few more:
213 is a person who's a bit stupid
1314 is forever, and of course 520 1314 is "I love you forever"
4242 is "yes, for sure" because 四二四二 (Sì'èrsì'èr) sounds a bit like 是啊是啊 (Shì a shì a)
and so on. Wikipedia has a list of common ones here. It's worth a read. Some are quite creative.
I'll write more soon on the best methods for learning. If you want to get a taste for it in the meantime though, my current favorite site is iTalki, and my favorite teacher by far is Amy He. If you decide to try it, click here and it's cheaper for both you and me.