That's a seriously tall order for a compact digital camera. Fundamentally compact digital cameras are not as good with low-light and motion because their sensors are much less light sensitive than the ones on DSLRs.
Some of this is just physics. The sensor on, say, a Canon Rebel XS (a DSLR) is about 22x14mm, and has 10 million pixels. The sensor on a Canon A480 (a compact) is 6x4.5mm and has the same 10 million pixels. Sensitivity of each pixel site is a function of the area. Do the math and you find that the area of the pixel sites on the XS is 12 times as large as the A480, which makes it roughly 12 times as sensitive to light. For the photographers out there, that's a 3.5 stop difference in fundamental sensitivity.
There are other factors, obviously; CCDs have different noise performance than CMOS sensors, there have been continuous improvements in sensors over the years, etc. But as a first rule of thumb, pixel area tells you a lot about quality.
To compensate, compact digital cameras do a few things: they crank up the gain on the sensor, they use much more aggressive noise filtering, and they try to use the low ISO ratings as much as possible. So in auto mode, it'll still be trying to use 100 or 200 ISO, because 400 and 800 will be noise-ridden messes. It'll try to use the flash, and if you force the flash off, it'll use a long shutter speed. The result is blur and dark and/or noisy pictures.
To really do well in a darker environment with motion, you have to use really high ISO ratings and good image stabilization (and you still might not get clear pictures; if it's super dark and there's a lot of motion no camera will do very well). Unfortunately, those are both characteristics primarily of DLSRs.
So after all that, if being able to slip the camera in your pocket is the most important factor, you still have some options to improve your odds of getting a decent picture.
- The current leaders for cameras with reasonable image quality at high-ish ISO ratings are the Canon G10 and the Panasonic Lumix LX3. Both are expensive; you could buy a DSLR for the same money. And they both use the tiny sensors that all compact cameras use (though slightly larger than the A480 mentioned above). But they do the best job of maximizing the image quality at high ISO, and if you're taking fast-motion or indoor pictures without a flash, you'll need high ISO.
- You may want to invest in a good noise-removal program. The one I like best is Noise Ninja, which allows you to profile the noise characteristics of your specific camera and apply just the minimum filtering necessary in each frequency band. It does a nice job, though it's not magic.
- Cover the noise with more noise. The noise produced by an over-amplified camera sensor isn't very aesthetically pleasing; it's more like TV static than like film grain. Get a plugin to add grain to the image, and it might help make the noise into an effect.
You could also just try to get the smallest DSLR you can find, with the smallest lens you think will be usable. The Panasonic G1 is a Micro Four-Thirds camera and is smaller than most DSLRs. The kit lens is still fairly large compared with the lenses on most compacts. It's only pocket-sized if you have a pretty large pocket. But you can fit it in a belt pouch easily enough.