I'm struggling with the same thing. I use Premier Pro and Photoshop and all the gimmicks those programs have to reduce the file size, but still it all usually leaves me pretty disappointed and the end result is not at all what I was looking for. Resizing the image is the most effective but if the animation is too long, let's say over 10 seconds, nothing seems to be enough.

Perhaps using only three or max. 6 colors in the art to begin with? Could also generate fun stuff, dunno. I feel your pain. Your pain is my pain. A gif headache.