I am currently a freshman in college and am possibly considering going to grad school for an OTD. Does it really matter what you major in as long as you complete the required/recommended courses?
Also, does the prestige of your undergrad school matter in OTD? Currently, I am at a state school, but was admitted to a top 20 school as a transfer. Is it worth it for me to transfer to the better, but much more expensive, school?

No, no, and no. As long as you complete the pre-reqs, your major really doesn't matter - however, they love a firm background in psychology. Plus, it would benefit you. Where you did your undergrad work doesn't matter, either. It also doesn't matter how expensive your school is - private institutions are much more expensive and really don't offer anything more. Find a public in-state institution with a good reputation for their OT school. Good luck!