There are several cameras capable of handling this kind of shooting on a consumer budget. If you're interested in purchasing a professional-quality camera that can change lenses and show you exactly what the lens sees through the viewfinder—what's called a DSLR—Nikon's D5000 is a good choice. It can be had for less than $500, then combined with a wide-aperture lens for $200. That package gives you the same low-light capability of cameras that cost thousands of dollars. If you're looking for a point-and-shoot, the $420 Samsung TL500 has the best high ISO capability currently available and combines that with the widest aperture expected to be on the market anytime soon. Several other point-and-shoots with aperture priority mode are already out or soon to arrive, and you can follow this link to compare them with the TL500.
Cheaper cameras can sometimes do a decent job, even without the ability to control aperture or the latest sensor technology: Try shutting off your flash and setting your ISO to 400, 800, or higher, and seeing how you like the results. But if you're eager to shoot better photos indoors or in other low-light situations, the added expense of a more-sensitive camera is worth it: When shooting with your built-in flash, there's little difference between a $50 camera and a $300 camera; it's only when shooting without a built-in flash that differences in camera quality really begin to reveal themselves. (For a few tips on how to make the best use of available light, click
One last advantage of shooting without your built-in flash is the ability to take many pictures quickly, often several per second. This leads to more experimentation with light, but also to better chances of getting "the shot": When I line up my parents with all their grandkids, it's impossible to get everyone looking and smiling at just the right time for me to push the shutter once and get the shot. Instead, I put my camera into "burst" mode and shoot perhaps a dozen different pictures, then choose the best one.
Shutting off your camera's built-in flash isn't a guarantee of great pictures, and there will still be times when a lack of light requires you to turn the flash on. There's no one magic formula, but shooting with available light is your first step in making your subjects more attractive—and also in making pictures that look like they're yours. Most professional photographers tend toward certain approaches to light that act as a sort of professional signature, so that viewers can look at a photo and say "that's a Gregory Heisler." By shooting without a built-in flash, you can begin to develop your own photographic signature, while earning accolades from friends and family for casting them in their best possible light.