Was everything supposed to revolve around carrier aircraft and submarines, or were the remaining gun-only cruisers and destroyers from WWII supposed to engage with their guns when carriers were not available? I noticed that the Harpoon Missile didn't even come into service until 1977, so there's a huge gap there where the only equipment a surface ship would have to engage the enemy was still it's guns.
U.S. Navy doctrine was to fight major "surface" engagements in Carrier Battle Groups. This is a fleet element based on one or more aircraft carriers for offensive strikes and cruisers and destroyers to defend the Carrier. Since a Carrier has a range of hundreds of miles, the main idea of a Carrier Battle Group is to stay at a very long range and sink enemy ships with the Carrier's air group.
Cruisers and Destroyers in this period were mostly outfitted with anti-air and anti-submarine weaponry, with most guns being "dual purpose" and also useful against aircraft. Since the Carrier would fight at ranges in the hundreds of miles, the Carrier's escorts anti-surface capability did not need to be extensive.
The answers provided that the US primarily on its carrier fleet for anti-surface work are true. However it is not the case that there was nothing US ships had but guns and torpedoes until the Harpoon. The first generation of US anti-aircraft missile (Talos, Terrier, Tartar) were modified in the 1960s to also serve in an anti-surface role, while the Standard missiles had that capability from the start. They were not as good as dedicated anti-ship missiles, but they were something.
This site, (which has a lot of background info about Talos missiles) has some after images of a live fire exercise of a Talos used against a test target ship.