I got a new-ish camera

It looks like you've got the ISO set at either 800 or 1600 for these images.  Almost all digital cameras will have problems when the ISO is set that high. The other thing you're probably up against, although there really isn't anything you can do about it, is that the sensor on that camera is pretty small.  That's going to affect the noise level as well.

When I was first starting, before I could afford lighting, I went to Home Depot and bought a couple of the cheapest clamp lights I could find along with a set of daylight bulbs.  They weren't great, but they did give a little more light.  Plus, being on all the time you get the chance to see what the light is doing before hitting the shutter.  You might also want to take a look at the Strobist blog.  The author does a really good job of explaining lighting setups done with minimal equipment.

Andrea, I am so glad you posted the second image; I looked at the first one and didn't notice the legs in the sofa!  I thought you'd posted an image of your child cut in half!!!  Just kidding, I did have to look at it twice though.

Oh, and just so I don't stop in here and not try to keep the thread on track; I agree that the ISO appears too high and the small sensor.  Small sensors are really difficult to control graininess (digital noise).

Travis

That depends on the camera and the type of "stabalization" being talked about.  I got that from the link that I provided earlier. 

Ed