Every type of shot that you just mentioned, night, fireworks and indoor are all likely to be blurry on any camera with your current technique.
All of those types of shots, including indoor in poor light (just because it looks bright to your eyes, doesn't mean the camera sees it as bright) need to be handled a little differently, they need to be still. A tripod for the outdoor shots is the best way to handle this. For the indoor shots, turn on some more lights, use a tripod (for "still" images, a tripod won't help when you're trying to shoot a 3 year old running around) or use a flash. Other things to try is to not zoom in but get physically closer to what you're trying to capture. The glass on both of those camera's is very similar; f/3.1-5.6 on the Canon and f/3.6-5.6 on the Sony. The Canon DOES have stabilization built into it, but it only works for certain types of images; a 3 year old running around for example, IS\VR\OS will not help.
I really don't think you will be a happy with a new camera, it's not going to change the situation.