If you are having difficulty holding your camera straight, purchase a tripod. A tripod will go a long way in helping to keep your camera in place, so you can focus on other variables other than balance. Tripods work great if you are in the wilderness or on an uneven terrain.
To avoid red eyes in your photographs, make sure that your camera either has built in red eye reduction or you change the direction of your flash. If you did shoot a photo and it contains red eyes, you can easily remove them through the use of a graphics software such as Corel or Photoshop.
A good photography tip that can help you is to take a step back from your work sometimes. A picture you might not have like before might suddenly seem appealing to you. It's very important to step back from your work so that you can always have a fresh perspective.