You start by downloading the Acrobat SDK.
In the SDK is information about developing a plugin for Reader, which will require you to obtain a license from Adobe. Submit the necessary paperwork to see if you are approved. It doesn’t make much sense to start development if you don’t get approval.
Once approved, you will want a copy of Adobe Acrobat for development as it’s easier than developing with Reader.
Plugins are developed in C/C++ and there are samples in the SDK.