In Need of Some Photography Expertise

I got a brand new Nikon Coolpix S8100 for Christmas, and have been having a good time developing my skills and learning the camera's features. I have been having some difficulty, which I am more inclined to believe is the result of human error rather than any shortcomings from the camera. I have learned in researching cameras that there is a whole language to photography, and I don't speak it.

The Nikon Coolpix S8100 is a point-and-shoot camera that says it boasts a lot of features available on a DSLR, but without the expense, the equipment, or the need of a skilled photographer. I am an amateur with no aspirations of doing anything more than taking the best possible pictures of my family. I do not expect my photos to be of the same quality of a DSLR or for my photography skills to even come close to such results. Just wanna put that out there so that the photographers and photography buffs who I am desperately seeking guidance from will not be insulted thinking I'm trying to create photos as beautiful as theirs with a point-and-shoot. Not the case.

In case this information is helpful, the Nikon has a 10x optical zoom, and allows me to select features such as: night landscape, night portrait, backlighting, indoor/party, and auto mode. I have been experimenting with the different features, with varied results. I promise to be brief, but if ANYBODY has a suggestion, I would be so grateful. If you know what it's like to try and capture an adorable moment with a baby, only to have the picture turn out poorly and the moment is now over, you understand my frustration.

This is what I would consider to be a "good" picture. At least, good for me. I use the Night Landscape option, I think it makes a nice warm glow in the picture. The downside is that if the subject is in motion, the image will be blurry. If you look closely, you'll notice that the image on Robinson's shirt is slightly blurred because he isn't perfectly still. This is a big problem with babies, not being still.

 This is what I get when I use Night Landscape and my subject(s) are in motion.

 Night Portrait is surprisingly worse, given its name.

I believe this is Auto Mode or Indoor/Party, without the flash. The camera features a flashing red light that flickers before taking the picture. It's to prevent red-eye, but I call it the "smile killer", as this was supposed to be a moment capturing Robinson's adorable smile. Until the red light flickered and he made this face. I imagine as he gets older, this will be less of a problem. Another thing I don't understand: clearly, this is a series of photos taken at the same time, with the same external factors. So why is one picture so much blurrier than the other? I may have chosen a different feature, like Subject Tracking, in the second photo.

 I try to use the optical zoom, rather than holding the camera closer to the subjects, but sometimes it's still overexposed. Sometimes, a bright flash in a baby's face is a bad thing.

This photo was either Subject Tracking or Auto Mode (if it's important to know which, I can take more pictures, but honestly they all turn out this way.) There must be some kind of shutter speed setting that I can adjust in the camera so that this won't happen.

I purposefully chose a high-end point and shoot so I could take beautiful pictures without needing to make a big investment. Now, I'm having to take 40 blurry, over-exposed pictures in order to get one good picture. I know I'm doing something wrong here, and I'm feeling frustrated and overwhelmed. Help me!