China is certainly much cheaper than the West to live in. I think for many people, that is one of the positive aspects of moving to China, as the potential to save money can mean you can return home with a large chunk of change.
Teaching English can provide a good salary, but there are also other opportunities to work on the side to earn more money. Private tutoring is a popular way to earn additional money that I know many people do. There are jobs to tutor online (if you are American or Canadian, VIPKID is a popular website to use in China), or there is in person tutoring. It is illegal to work on the side in a private tutor center, but you can easily be employed by parents and meet the students at their homes a few times a week. Depending on how long each lesson is and how many students are in each lesson, you can make anywhere from 200-600 RMB per hour. Keep in mind, this varies, and also depends on what the parent(s) are willing to offer.
Since it is fairly cheap to live in China, it can be easier to save money here than it would be back home. Keep in mind, how much you save will depend on where in Shenzhen you live, how often you go out drinking, and how often you eat take-out food (or more specifically, Western take-out food), and your other spending habits (since everything is available on your phone in China it can be easier to spend money). Rent in the pricey areas of Shenzhen for a shared bedroom apartment can be around 5000-7000 RMB per month, and depending if your company gives you a housing allowance or not, that can also take a chunk of money from your savings. If your company does offer you a housing allowance, it will usually be between 2500-3000 RMB, which is enough to cover half the rent on a two-bedroom place!
One difficult thing in Shenzhen can be banking and knowing what your account balance is. My company set up my bank account through ICBC, and for the first half a year of living in China I had no way of knowing what my current account balance was. You can always go into the bank to get a statement, however if the bank is not close to your school or apartment, this can easily become a hassle. I know many people have a text program set up where every time money comes out their bank they get a text message with their new balance, and I lucked out and got an app where I can see how much I spend and how much I have (although no one else seems to have this app except me). Overall, it can be easy to make additional money in China, and as long as you are wise about it, it’s not hard to keep that money and let it accumulate over time.