Are you open to the possibility of getting yourself a flash as a quick before-Christmas gift to yourself? In my experience — and especially back when I had young, enthusiastic children not interested in sitting still — adding a flash was the best thing I did for my family-around-the-Christmas-tree photos.

On-camera flash is annoying and doesn't provide good light, but for under $200, you can get a radio-controlled flash system. Then, put the flash on a bookshelf or somewhere in the corner of the room where it bounces from the ceiling. Take a few test shots to make sure the shadows aren't crazy from this, and put your camera's shutter at the X-sync speed (¹⁄₂₀₀th, for your camera), and use f/5.6 or f/8 to get good depth of field (more of everything in focus). Then, experiment with flash power and ISO (start around 400 or 800) to get the exposure right. Once you have this, you won't have to change any settings, since you control the light.

Without a flash, this is generally a very difficult situation, since indoor lighting is generally much dimmer than people assume it is (our brain does a great job of adjusting), and the only way to get in enough light is to either leave the shutter speed open for a long exposure (risking motion blur) or to use a fast lens with a wide aperture (f/2.8 or faster), which means not much depth of field. Or you an use the very high ISO modern cameras offer, which will amplify the signal and produce noisy results (adequate for social sharing, maybe, but if that's all you wanted, you could just use your phone camera).

If you're doing formal "stand in front of the tree" pictures, it's a little easier, but even then, the flash will really help.