Not an expert, but I had 24 hours in Busan after coming off the hydrofoil from Japan and before continuing on to Tokyo, and I decided that the most efficient use of my time was to take a bus tour from one of the companies that leaves from the front of Busan Station. They had two circuits, one that goes around the central city and another that goes to Haeundae.

It was a quick once-over, and we saw things entirely from the windows of the bus, but if I ever go back, I will have some ideas of what to see and do. Busan struck me as being an intriguing combination of Old Asia and contemporary, futuristic Asia.

And don't forget--there is always the option of buying a guidebook and reading up on the area. If paper books are too retro for you, many of the standard guidebooks are also available as e-books.