You have three choices with low light photography. Focus on the subject, depth of field, and clarity of the photo (amount of light collected by the camera). But you don't get to choose all 3. If you're lucky, you can get 2 out of 3 if the subject isn't moving.
If the subject is moving at all in low light, you cannot get a clear picture with any camera without a flash. You have to choose what you want to sacrifice in those moments.
A couple of things that I have found helpful for low light pics:
--Long press and hold focus on the subject in auto mode (this also adjusts the exposure to the subject, which helps if there is a bright light source in the background like a TV)
--Turn on the flash, and adjust the exposure to -0.5 or -1.0. This will reduce the amount of "white bright" flash blast the subject gets. You can also increase the exposure if the subject is farther away from the camera and the standard flash setting doesn't illuminate enough. The general RoT (rule of thumb) for adjusting the strength of the flash: The closer the subject, the less flash you need, the farther away, the more you need.
--Use a burst shot. Hold down the shutter button in Auto Mode (or one of the volume keys) and the camera will take up to 30 shots of the subject in a matter of a few seconds. This is great for moving subjects, but you may not get a great shot in low light because the shutter speed will be faster (resulting in less light capture).
It helps to play around with this camera a little bit. When I first got it, I didn't think it was very intuitive. Once I realized how it worked, I have had no trouble getting great shots, even in a "point and shoot" mode.
And don't be afraid to do an old photographer's trick. "Back in the old days" when we actually had to develop the film before we could find out if we got the picture we wanted, a good photographer would "bracket" his shots. That is, he would set f-stop and shutter speed to what he thought the shot required, then do a couple of shots on either side of that setting, just to make sure he caught a good picture. I still treat digital cameras like SLRs. I take 5-10 shots of just about anything I am shooting, in different modes, angles, exposures, etc. With sports or other action shots, burst is my friend. I often sift through 50-100 pictures from one event to settle on 10-15 for an album to share with friends. Most of my best shots came from using this method. Among other things, it makes your subjects less "aware" of the camera so they start to relax, which results in better pictures over time.