"At the weekend" is a British English usage. "On the weekend" is an American English usage. "On the beach" means you are literally on the sand near the water. "At the beach" is more general and could mean you're literally on the sand (or in the water) or it could mean you're visiting a beach town.

I think it's confusing for most people learning English. THERE ARE NO RULES. If all you want to do is to have a conversation with someone in English, errors in choosing prepositions are not important and can be understood. In U.S. usage we would use "on weekends" to express a habit or custom. "On weekends I go to the movies." "On the weekend" is correct but it's probably not what we'd say. We'd say "What did you do last weekend," "What are you doing next weekend," "What are your plans for the weekend." It is just as Susan612 says: "We go swimming at the beach." "There is a shabby little amusement park at the beach." "I like to eat fried clams at the beach." "I have an umbrella I can set up on the beach." "We can play volleyball on the beach." "We spread out our beach towels, sit down, and sunbathe on the beach." "Kids build sandcastles on the beach." "The Allies landed on the beach at Normandy during the Second World War." There is a specialized use of "on the beach:" people in the British Navy are "on the beach" when they are retired or between voyages.