What program is best for turning a childrens book pages into simple animations?

Adobe Character Animator is very easy to use to animate cartoon characters.  It can create lip sync from audio files or translate your face and head movements via webcam.  I use it a lot to create these animations, which I then bring into After Effects to animate against backgrounds.

With After Effects, I think you'd get better results in the long term, but there's a lot to learn before you'd be able to make a start for a project like there, whereas with Character Animator, you'd get results very quickly, though evenutally you may decide you want more control (which AE will give you)

Storyboarding is usually the first step in creating an animated movie. The illustrated pages of your book probably take care of that part of the development process. That gives you the essential scenes you will have in your final movie. It might even give you the individual shots you need for each scene.

The next step in the development process is figuring out how the characters and props need to move in each shot.
In the good old days, animators did pencil sketches showing the characters' starting, middle, and ending positions. Then the in-between frames were drawn, and a Pencil Test was filmed (rendered) to ensure the shot worked. When the Pencil Test was OK, the sketches were sent to the Ink and Paint Department to receive their final polish and color. Then background plates were added to complete the shot, and the animation cameras went to work creating each shot. 

The final movie was edited with sound and music when the shots were all processed.  

With a computer-based approach, you need to start with layered PSD or AE (Photoshop or Illustrator) files with every part of the drawing you want to animate on a separate layer. By far, the simplest approach would be to use Adobe Animate. Step 2 would be to add After Effects to the mix. The final step would be to render the shots you create using Animate or After Effects, or Animate + After Effects, to a suitable lossless 10-bit or better production format. The rendered shots are then taken those shots into Premiere Pro for final editing, audio mix, music, audio effects, and even the final color grade. That basic workflow is how all professionally produced animated films are made. 

That same workflow is still efficient. Get the characters moving the way you want them to move with few effects, even skipping frames or running previews at lower resolution to make sure the scene works, then add effects, motion blur, lighting, and backgrounds, then check a few of the most critical frames, and then render the shot.

I hope this helps. When you have the artwork separated into layers, it will probably take you a couple of days to get a decent handle on how to use Adobe Animate. Adobe After Effects has a longer learning curve. It will probably take you 30 to 40 hours to understand the UI, workflow, and techniques in After Effects. If you want your characters to turn from a left profile to straight on to a right 3/4 profile, you will need to generate a bunch of in-between poses for the characters. The time required to do that work depends on your Photoshop or Illustrator skills. 

Another option would be to create your characters in a 3D app (Blender is a free and plenty capable 3D app used by a lot of professionals). You are probably looking at about 60 to 80 hours of study and experimenting to get proficient enough to be at a semi-professional level with any 3D app. The same Pencil Test, Ink And Paint, Edit workflow applies to 3D apps. 

Hi Christine,

I love working with After Effects when it comes to character animation. We'd be able to give you a better suggestion if you could share a small clip of your animation.

Let us know.

Thanks,

Nishu

After Effects is a good option to do that, but it's not easy at all if you're new here, it's better if you get some ready template (and there is a lot) and customized it to animate your book
a few examples from videohive https://videohive.net/category/after-effects-project-files?term=book%20animation#content