I dont agree with people saying shore vs. beach means they are a benny. Yes, you should say beach but saying you are "going to the shore" basically means you are actually going to the beach at the edge of the shore/shoreline.

How about when I went to the "North Shore" to pipe, sunset and waimea, sounds same to me. Anyway, I guess people refer to "going down the shore" ever since signs went up welcoming people to the jersey shore.--but who knows.