I've been learning how to animate fractals I made in Apophysis.
I learned two ways to create the frames/images in preparation for animation.
1. Use Apophysis scripts that come with the program
(There is a script to preview and a script to create the frames.)
2. Use a script that uses Flam3 rendering for a different animating effect.
Those scripts took some study and work before I could use them.
Once I had created the frames (images) then I needed animation software to animate.
I used Paint Shop Pro 9's Animation Shop. Frame transition effects, like fade, in Ani Shop is another option to try.
Part1: Beginning Scripts and Animation
UPDATE 22 Apr 11: Dead link. Look for info here.
Claire Jones has tweaked the original Apo scripts.
Flam3 Script Tut
VirtualDub was the recommended free software to create animations, but I have not tried it.