In my contract, it states that I must give six weeks notice when I decide to terminate my contact. Is this legal in California? I've never heard of a notice period that length. Also if I signed the contract but never received the signed copy back from them, can they enforce this clause?

Just because you don't have a copy of it, doesn't mean it's not enforceable. You signed it, whether or not the provision is common or not. Why do you think it isn't valid? Honestly, you will have to get a copy and have an attorney review it anyway. There's no way we can interpret a document we have not read in its entirety. However, depending on your position, 6 weeks could be very reasonable.