Three basic scams to watch out for. Not that these will necessarily make or break your trip but it's probably a bad sign. The smart money says just don't take the risk when confronted by any of these things.
1. "How quickly can you get here?" Not proof positive that bad things are about to happen but if they're trying to rush you, like any relationship someone too desperate who doesn't care who you are can be a red flag for two reasons.
One is that this tactic may be used to pressure you and get you to rush on over before you've had a chance to think things through.
Two, it's also a bad sign because it may mean the previous teacher left in a hurry without notice. This may not be the schools fault but few of the employers I go on interviews for back here in the states care enough to give me the benefit of the doubt. It is important enough for you to have at least a decent experience and there are enough schools in China that you can be a little picky. Make sure a school treats you right and doesn't ask you to hop on a plane after the first date.
2. "Let me hold that for you." This should go without saying. Never let your boss 'hold on' to your passport. Without it you kinda can't leave or have difficulty doing so at the least. Obviously they will try to hold it so you don't run away from them when you realize how horrible the school is. You may need to hand your passport over to get visas and other documents. Make sure you get it back and in a reasonable amount of time. A slightly related scam is they'll tell you to come here with a tourist visa and we'll get you a residence permit which may or may not ever happen. If they can legally get you a residence permit they should be able to get you the Z visa before you go there.
3. "I hate you, I hate you, I hate you" Another scam I've heard is that they'll try to force you out around the nine month mark or even right up to 10- 11 months. This is because after 12 months they have to give you another 7k-10k RMB for your return flight, but if you choose to leave because they're treating you badly or they are now all of a sudden unhappy with you then they save some money and they were bringing someone new in shortly anyway. Sucks to get that far and have a bad time. One advice is mind your p's and q's, don't give anyone any excuse and be strong for a little longer. You could possibly even take the hint and go. You might lose some money (and feel like a failure) but you're not leaving yourself vulnerable to who knows what.
This could jeapordize your contract though too, idk how legal these contracts are and if yours has penalties if you leave before a year. I'll say that it seemed like then, and still seems now that you are worth enough and have enough prospects for work that you can go without a contract. Unless there's something really wrong with you and you have a bad attidtude from the start you'll likely be fine. You don't need a contract securing your right to work. Even if you did have a contract and you turned out to be that bad, I'm sure any sufficiently creative boss can and will get rid of you anyway, contract or not.
Skip the contract completely, don't mention it, if they bring it up tell them you'd rather not have one, or write your own contract if it is required by the state for a visa or RP. Keep it simple, "i work for you and you pay me this much each month." If they want to push you around later and renegotiate our verbal contract then you are free to walk without worrying too much about your lack of knowledge on Chinese law.
They will still want you to have a decent enough apartment that will keep you there, they will still have to pay you to keep you teaching. There's not much in your contract besides shelter and payment anyway, maybe work hours but again, we're adults we can reach a reasonable agreement and if you're unreasonable... well i guess with Chinese visa laws you can't just go somewhere else to work, even though there's plenty places, not without a release paper but you can go home. I don't know how legal these contracts are anyway nor if one is required for the residence permit. Going or staying both have some risks if someone is trying to push you out after 9 months it's because they don't want to pay you.