You focused on the nose, and generally you want to have the eyes as the focal point. Also, not many people can make the on-camera flash look good as it's very white (and it's not having a good effect on the eyes). Try using natural lighting to soften the appearance and take the contrast down a bit.
When dealing with pets it's easier to get a good composition when your on their level. Pop-up flashes are especially harsh on short hair animals.
The best thing you can do is read a book. Most people recommend "Understanding Exposure". I also liked "Digital Photography Simplified". It's a quick read packed with useful information.