Does my contract with rent a center go to a collection company if I don't pay?

That is up to the rent a center.  Barring something in the contract which expressly prohibits it, rent a center can hire a collection agency to try to collect on the debt it claims, or it can sell the contract to another person or entity who will then try to collect.  You do have certain rights with regard to unfair collection practices, however, most of which are codified in the Fair Debt Collection Practices Act, a federal statute.