Kanagawa prefecture has the second largest population in Japan and its economic activity is much the same as Korea.
Around the Miura Peninsula, Keikyu has a practical monopoly on transport, with its trains and buses being cheaper and more extensive than JR. They offer one-day passes which are good deals if you want to explore the area (as well as Tokyo area passes for travelers from Yokohama). Inquire at Shinagawa or Yokohama station for details.
For Enoshima and Kamakura, JR Tokaido Line(東海道線), Yokosuka Line(横須賀線), and Shonan Shinjuku Line(湘南新宿ライン) is a convenient way. Narita Express Train has direct service to Ofuna Station, two stations from Kamakura Station.
Kanagawa has a great variety of outdoor activities available. Hiking, mountaineering, sea kayaking, rock climbing, para-gliding, and much more. Kanto Mountain Adventures leads rock climbing, hiking trips, and more in the area for reasonable prices. There are many great hikes easily accessible by train, especially the Odakyu line.
Hiking is available around Honatsugi, Isehara, Shibusawa, and Hadano in the nearby Tanzawa Mountains. The Hakone Mountains also offers plenty of hiking oportunities.
If you like beaches, Enoshima is the most popular one, and gets very crowded in the summer time. Windsurfing is a popular activity there, as the waves are rarely big enough for regular surfing due to the surrounding landmass. Enoshima itself is the island right next to the beach connected by a 600 meter bridge. There is a tower at the top of the island, where you can get a view of Mt. Fuji, the ocean, Oshima, and the Shonan Coast (the stretch of beaches in Kanagawa). To the west of Enoshima, across the Katase River, is Chigasaki Beach, which is also very famous. It is a small, but relaxing beach. There is a big symbol "C" for Chigasaki, which is a popular picture taking spot. In August, they have a Fireworks Festival.