You could use Macromedia Flash MX to make moving GIF images. What did you have in mind? It will do the PNG ones to...I'm only saying Macromedia because thats what I have.
I don't know what other programs are out there that can do animated stuff.
You need an image editing program like Photoshop or GIMP. GIMP is better, because it's free. A simple internet search should tell you everything else you need to know.
But a better option than animated .GIFs is animated .PNGs, which you only need the APNG edit Firefox extension for to do. I strongly recommend animated .PNGs over .GIFs, they're so much better.
If you don't like Firefox there are other programs that do Animated pngs. Kyria does use firefox but doesn't like the addin for making Apngs. So she uses japng-editor, a nice little java script apng assembly tool. it's freeware.
As Togan said really only one browser out of the many seems to show them correctly. SO if you do use them, you'll just have to be prepared for a lot of people not being able to see aything beyond the first frame.