Photographic Gear Needed
Camera Body Any digital SLR camera should work just fine for surf photography. Surfers are usually in the water when the sun provides enough light where you don't to raise your ISO to the point where noise becomes a problem (unlike night time football). Advantages a higher-end camera will have are the faster frame rates (being able to shoot at 5 or more frames per second is recommended), weather and sand proof bodies, and more advanced auto focusing systems for faster and more accurate focusing. That said, any SLR should be able to meet the challenge since we will be shooting in a decent amount of light (even during dawn and dusk).
You should have a normal zoom lens (24-70mm or similar equivalent) to capture wide shots of the beach and surfers and a telephoto (300mm or more) to get up close to the action. The longer and wider aperture the lens has, the closer you will be able to get while blurring the surfer's background. Be warned, these lenses can get very expensive!
Photo by mikebaird using a 300mm F/2.8L IS USM lens on a Canon 5D
A monopod and ball head might help to reduce the stress on your arms depending on how long you plan on shooting. A tripod will probably get in the way and slow you down.
Using a Point and Shoot
Using a point and shoot camera is doable. Most consumer compacts do not offer enough zoom to get close enough to your subject so you are limited to wider shots. This can be made more interesting by standing further back on the beach and shooting with something in the foreground (child playing in the sand, the beach, onlookers, etc.) These foreground elements help set the scene and/or help tell a story with your photo. If you have the opportunity to go out on a pier near where surfers are surfing, be sure to capture the beach, surfers, and waves.
If you happen to have a waterproof point and shoot, you're in luck. Use the waterproofing to get up close to surfers and achieve a different perspective. Try getting down low into the water (or even underwater) for unique shots. If you are a surfer, you can also try some "first person" perspectives with your surfboard as you ride the waves.
Other ultra-zoom point and shoot cameras have longer lenses. These can help you get up close to surfers, but these lenses are not as sharp as those in an SLR. To make things more difficult, you are also really prone to camera shake when zoomed out so far and there is a significant delay from the time you press the shutter till when a photo is actually taken. An SLR with a better lens will help to mitigate these issues.