If you are near Haeundae, then Jangsan mountain can provide a good day's hiking. There are military bases at the summit complete with minefields (Clearly marked and fenced off), and great views over Busan and on a clear day to Japan. A hike to the summit and back should take 3-4 hours. Hiking trails are not really well marked, even if you can read Korean script.
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주천조각보박물관
Museum of Korean traditional art.
성불사
Buddhist temple and monastery located up Jangsan mountain. Good scenic views over Haeundae.
An old mainline trainline has been repurposed into two types of tourist train - a 'sky capsule' (an elevated small tram) as well a coastal train.
해운대
One of the most popular summer destinations in South Korea. Haeundae beach attracts tourists from all around the country and gets really overcrowded in late July and early August. Deck chairs and parasols are available to rent for ₩10,000. Hotels including international chains are nearby, and the Busan Aquarium is on the beach front. On a clear day, Tsushima Island (Japan) can be seen. In winter the beach is much more quiet, although various events such as the Pusan Film Festival and the new year Polar Bear Swim are held here. Away from the beach in Haeundae town there are some excellent restaurants and a wide variety of shops, although many are hidden away. Explore the streets behind the beach front boulevard. The Haeundae Tourist Information Center is directly on the beach. Surfing and jetskiing is available, although most people seem happy to float in yellow inflatable rings (they can be rented).
부산 아쿠아리움
This is the largest aquarium in Korea with great facilities and excellent displays. There is also a 3D ride that is worth checking out. There are larger aquariums in Asia, and outside the facility looks rather underwhelming, but inside the displays are high quality and the tanks well maintained. Easy to spend a couple of hours with kids. The price is ₩23,000 but you can get various discount like birthday dc.
This boat trip starts off at Haeundae’s Mipo Ferry Terminal and heads towards Oryuk-do Islets and the Igidae area. The shores of Igidae are composed of breathtaking views of fantastically shaped cliffs. This 1-hour boat trip offers great views of Busan Harbor. The boat ride can be very bumpy and there is no way off, so check the weather conditions if you feel sea sick easily!