Also, it sounds like you're complaining about the flash... it's two different things to get good low-light performance
with the flash and
without the flash.
To get good performance without the flash, you need a big sensor so that you can use a higher ISO without the picture turning into a noise-filled mess. (Or a faster lens, if you move to a DSLR.) DSLRs have much, much larger sensors than any point-n-shoot digital camera, and if you stick to PnS cameras, the big-sensor Fujis (like the F31 with 3x zoom or the S6000fd with 10x zoom) are the clear kings.
To get good performance with the flash, you pretty much need a standalone flash unit that you attach to the top of your camera. To do this, your camera needs a "hot shoe" - all DSLRs have this, not all PnSs do. The bigger flash will be able to produce a brighter flash. Ideally, you'll be able to bounce this off something - like in a normal house, you can point the flash at a 45' angle to bounce off a white (or at least, brightly-colored) ceiling and the resulting image will look much more natural and less washed-out. If you're outdoors or something with a very high or black ceiling, you can't bounce, so you can try a diffuser, which will soften the flash's effect to make it less harsh. I haven't played with a diffuser so someone else would have to tell you how well they work in the real world.
The Fujis are pretty cheap ($200 for the F31 and $250 for the S6000fd, both after rebate), but a DSLR can get a little pricey. Even the best deal at the moment, equipped for low-light shooting, will be several times more - almost $900 for the Pentax K100D with 18-55mm lens, 50-200mm lens, and a 50mm F1.4 lens which does incredible things in low light. You could save $150 or so by dropping the 50-200mm, but you wouldn't have much zoom available for the times that you did want it. You could also skip the 50mm F1.4 and still have better low-light ability than any PnS, but the 50mm takes it to a whole different level.
For camcorders... you're talking to the wrong guy.

I think most of them can do fairly well in low light nowadays, within reason? However, if you're looking at a high-def camcorder, there goes most of your budget right there, and I'm not sure how low-light performance will be. The more information you try to squeeze out of an imaging sensor, the worse the noise levels will be, but I'd assume that high-def camcorders have larger sensors than the regular standard-def ones, in order to overcome this. But I'm not sure, especially on the cheaper ones.