To draw and animate them I use Pencil2D and to assemble them into an animation I use PhotoScape. Both are free software. Pencil2D can output as an animated gif, but I've found it's very important to adjust the timing of each frame, speed up certain ones, pause others. This makes things look much more natural. At first, I usually just drew things and then wanted to finish up. I put little effort into adjusting the timing. The results were not great. Now, I really try to adjust the timing and things come out better.
Sometimes, I've also used Paint.net and RealWorld Paint (also both free). I've used these when I wanted an animation that has transparency and end up with an apng file (rather than a gif that does not have transparency). The heart with LUV is a gif with a white background. The heart without letters is an apng and is transparent. (Not sure why it's not pulsing, at least not for me as a draft right now.) !LUV
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