Down the shore is the term used by people from New Jersey - specifically Northern Jersey (away from the shore area) - to describe going to the beach areas. Since New Jersey's beaches are all in the central and southern parts of the state rather than the Northern part of the state, which is blocked by New York City so there are no beaches, going "down the shore" means you're heading south. That's where the "down" part comes from. People who live near the beach wouldn't say "down the shore" because they're already living in that area. Once at the shore, if you're going to the beach you would just say "I'm going to the beach". But the term "down the shore" actually just refers to heading south towards the shore points on the New Jersey coast, or to the general shore area, because the Jersey Shore is more than beaches. It is also made up of boardwalks (with rides, food, water parks), bays, lagoons, many restaurants, etc. So if you are just going to hang out in those areas, you would probably just say "down the shore".
Girl from North Jersey: I'm going down the shore
for July Fourth weekend!
Guy from North Jersey: Cool! What are you guys planning on doing down there?
Girl from North Jersey: You know, go to the beach, hang out on the boardwalk, go on some rides. The usual stuff.