Make sure your clay is completely dry before firing it. It could crack or explode if you do not.
When carving designs into clay, wait until the clay is "leather hard" -- dry enough that it feels like leather. Also, don't "scratch" by using deep, thin marks. Make your carving wide enough for the depth you are carving.
If working on a piece over a few days, keep the piece covered in plastic overnight, preventing it from drying too fast.
An easy way to make small animals is making smaller spheres and attaching them, then smoothing out the joints.
Clay is forgiving, but it gets tired and frustrating to work with after too much exposure to water or too much manipulating.
Always make sure the clay is completely dry before you fire it. Damp clay turns to steam, which makes your pot explode as it tries to escape the clay.
Ideally, you want to find someone who has at least a little experience to teach you. This is a very hands on process, and having someone to show you and interact with you is very valuable. This "how-to" is meant as a reminder or a rough guide, the actual hand positions vary with every artist.
Sometimes local colleges will give you enough clay to let you play a little. They might even let you use their studio.