I just picked up my first ever prescription for ocps. The insert says that I should wait until I have begun menstruating to take them. Is that true?

Yes. Make sure you follow the instructions for best results. In general, what you find in the package insert is what was proven to the fda upon submission of the medication for use by the public. It is what is scientifically proven & reproducible.
Maybe not. If you start the pill when you start your period, you are less likely to have breakthrough bleeding and you have contraception from the beginning. If you are having regular periods, that is easy. If your periods are irregular, go ahead and start the pills and use something else for birth control for the first two weeks. You may have irregular bleeding, stay with the pill - it will work out.