If I factory reset my iPhone would it mess up my SIM contract? I have an iPhone 6 and would like to know if I could factory reset my phone and still be able to use all my data, minutes, and texts as soon as I set the phone up again?
Typically a reset does not affect your contract, because the iPhone's state is independent of the SIM contract. Although it is best to verify that with your SIM provider, you should not need to worry about the contract falling apart because it is bound to the physical card itself.

Let me explain:
That means that while a factory reset of the iPhone may sound like every piece of information linking back to you is permanently deleted, some data like the Serial Number, MAC address etc. is still unique to the device (hence doesn't change upon reset).

As logical links such as those created by the software (iOS) are prone to change, physical ones such as the unique identification mentioned above don't. If, and yes, only if the provider of the contract linked the SIM to your iPhone, it would most certainly be linked to the unique codes such as your iPhone's serial number - for greater accuracy.

The iPhone (or any phone for that matter) does not track your usage, the network provider does.