Depends on how much you want to spend, and what are you looking to do. Personally I have 3 cameras I take with me. I have a dedicated JVC video camera, Sea Life waterproof camera that double as a point and shoot, and a Nikon D3000 DSLR with a external flash and 18-55, and 70-200mm lens. With the Nikon I can take pictures in just about situation, and any distance. The Sea Life works great for keeping in my pocket walking around the ship, and in its housing, can use it in the ocean pools, etc. The video camera does great in just about any situation also.
Heres the differences in some of the cameras out there.
Point and shoots, are good for spur of the moment pictures or walking around and finding a character in an unlikely spot. They normally come with a digital zoom, but the digital zooms past a certain point tend to distort the pictures.
DSLRs like the Nikon D3100, can do a lot more, but they come with a cost and weight. They typically are in the 7-800 dollar range for a good setup. But heres the big advantages. You can change out the lenses based on what you are trying to do. For example, I have a 18-55mm lens that has vibration reduction and image stabilization. This lens works great for close up work etc. like character greets, general pictures, things like that. The 70-200mm lens, I can reach out and touch a subject. You cant do that with most point and shoots. With the external flash, I can take somewhat professional pictures in any condition. After a certain point, the point and shoots wont be able to provide enough light.
I'm not advocating that you go out and buy a DSLR setup. Just pointing out some differences. Cameras are simply one of those preference things. I like to play with pictures and cameras. So the extra weight of my DSLR doesn't bother me at all. If your going to be using a camera for vacations only, point and shoots might be a better option. Olympus and several other manufacturers have cameras that are drop proofed, and also water proof up to a certain depth. They also take great video.
Whether its a DSLR or a point and shoot, the cameras all have either micro sd cards or sd card slots, and almost all you can get extra batteries for or carry spare AA or AAA, or the appropriate sized spare. On my cameras, using 2 and 4 gb sd cards I average between 500 and 1200 pictures and video capability. A lot of it also depends on what your camera settigns are but that's a whole different ball of wax.