I have pretty much survived on Jigga (that), nihao (Hello), Chi Chi ni (thank you) and Sigh Gen (Goodbye) (note: this is not the correct spelling, but the words are written frenetically so you can pronounce) since I have got to China. Not because I am too lazy to learn, but because I have not started lessons yet. However, today I successfully brought a sim card from a shop with only using these words. This may seem rude to you, however the people are very pleasant, and as long as you thank them (chi chi) and be respectful they will stay that way.
There are apparently two Sim card companies in China; China Mobile and China Unicom. I have not seem China Unicom in Zhuhai yet, as everyone seems to use China Mobile. You can buy a sim card from pretty much any shop you see. As you walk in the will have a glass cabinet at the till, in this there will be a series of numbers in English. These are the phone numbers for each sim card. You will need to pick a number and the price is dependent on this. If you choose a number with the most amount of 4s, your sim card will be very cheap, as 4s are unlucky in China.
For my Sim card I paid 65 RMB, but have paid 50 RMB for others, it really does depend on the numbers. Point to the number you want say Jigga and they will pick out the sim card for you.
They should put the sim card in your phone automatically- this is important- you need them to set the phone up for you so that you know that it actually works. There are a lot of scams like this in China, and being a foreigner you are the perfect scamee.
Your phone number will be on the back of the packaging, so make sure they do not throw it all away. Make sure you say Chi Chi Ne (thank you) and you should be ready to go!