This is a lovely place to go if you want to enjoy the beach. It's also beautiful when its not bathing season. Go check out these little beach houses in beautiful colors. Also make sure to go for a walk in the forrest just next to the beach.
Ordrupgaard is a beautiful museum with lots of interesting exhibitions. Right next to the museum Finn jul's house is located, a famous danish designer and architect.
Whenever I feel uninspired I visit the (now genericly named) Design Museum, formerly known as Kunstindustrimuseet. Sort of the V&A of Copenhagen. An old museum filled with gorgeus fabrics, weird artifacts and a cozy library filled with books about arts & crafts from around the world. If you're still uninspired after a visit here you're not able to. Don't forget to have lunch in the beautiful center garden surrounding by the overgrown buildings, statues and old trees.