After I take the ASVAB, what is my obligation to the military?

You have no obligation to join the military. However, you will be required to sign an authorization that permits the release of your scores to all of the military services. Then, you will undoubtedly be called by recruiters from the Army, Navy, Air Force, and Marine Corps, as well as the U.S. Coast Guard, so they can try to convince you to join their branch of the service.