In a case where local law and federal law conflict, which is the automatic winner?
There is no "automatic" winner.
If We the People have chosen to grant a specific power to the federal government, then the federal law applies. However, we have not granted total and absolute power to the federal government, therefore it does not retain authority in all areas of American law.
Of course, I'm speaking idealistically, because the federal government routinely exceeds its Constitutional authority and does whatever it wishes, without regard to the Constitution or the proper role of government.