I have created a set of text and video tutorials for The Animator and The API. I will create more as new features are added. Please contact me if any features are missing or you spot anything wrong! As far as I am aware, these are the only lessons for my software anyway on the web. If you have written your own and would like them to be included here, feel free to contact me.
The Animator is very graphical and requires visual demonstrations, so is best consumed in video format, hence there are only video tutorials for it. You can find them here on my YouTube channel. Videos take a while to make, so they might be slightly outdated. However, the concepts outlined in the videos are correct for all versions. The GUI might look slightly different in your version to the version used in the videos.
The API requires coding, so it makes sense to have text tutorials as well. The playlist of API videos can be found here. Each text tutorial accompanies a specific video, so I'll post a link to the respective video in each text tutorial. I have also posted the source code for the API tutorial online, you can find it here on GitHub.
At the moment, the API is only in Minecraft 1.7.10, so all these examples are for 1.7.10. Included in the list are the version of the API that is required, if you have the latest version you will be able to complete all the lessons. I will keep an archive of old examples if the API ever changes dramatically, but it should remain fairly consistent over different versions of Minecraft.