Anyway, for starters I would highly recommend some sort of footwear. Soft sand will conform to your foot a little, but hard, wet sand will not-it's not much softer than running on concrete, in fact (Don't believe me? Take a ladder to the beach, climb up, and jump off.) So shoes will support your arches, protect your toes, and cushion your knees.
As for what kind of shoe, I suppose the simple answer is any ol' running shoe. Then again, in recent years several companies have started making athletically inclined beach shoes that are built like running shoes but that have mesh and drain holes in case you get wet. One I like a lot is the Adidas Hellbender ($90). These shoes have a running-shoe footbed for comfortable striding, grippy outsoles for clambering over slippery rocks or logs, and a synthetic mesh upper that drains quickly and doesn't mind getting wet. They're very comfortable. Nike's ACG Air Hydrous ($65) is a little lighter than the Hellbender, using a featherweight material called Phylon for the midsole, and a neoprene and mesh upper for decent foot protection. A nice shoe, but best if your foot motion is very neutral, not slanted toward under- or over-pronation.
For something really different, try the Bite Headhunter ($70). It's basically a bulked-up sandal, with an over-the-toes front bumper for protection. This design has obvious advantages for beach use, but at the same time may allow sand to easily get between the straps and your skin, causing some major skin erosion.
So, try one of these. And happy running!