I would first advice you to download Microsoft GIF Animator. It's freeware, so try to find it on microsoft.com or a diff site. It's a really good program for beginners, works with frame by fame.
After that, you should download the Macromedia Flash trial. This programs is for more advanced animators. It's not really different that the GIF Animator from Microsoft when it comes to making simple sprite animations, it also works with frame by frame. But if you're going deeper with animating, then Flash could be a but more difficult than you think. Ask around if you want to know more about Flash, there are some good Flash animators here on MKO.
Still, first try Microsoft GIF animator before you start with flash. Play around with it and you'll be making animations in no time.