I'm afraid the answer is slightly more complicated than indicated here. First of all, you do indeed need the (free) Adobe Acrobat SDK, which you can download here: http://www.adobe.com/devnet/acrobat.html
This will give you the documentation and sample codes for a number of different approaches.
However, you need to realise that you will not automatically be able to write plug-ins for Adobe Reader. Adobe Acrobat Standard or Pro, yes. Adobe Reader, no.
Adobe Reader is a free product and Adobe allows only those plug-ins that wouldn't violate that concepts. It is for example not permitted to write a Reader plug-in that edits a PDF file and saves those changes. Those are things restricted to Acrobat. To enforce these limitations, Adobe makes you sign plug-ins before they will actually work in Reader. During the signing process you will have to tell them what you intend to do, sign a bunch of papers and fork out a sizeable amount of cash. And if they don't like your project, they might not give you permission at all.
When you download the documentation however, also look at what you can do with JavaScripts. Things like adding menu items and buttons are normally possible with a bit of JavaScript that is executed as Adobe Reader launches. Once again though the functionality you'll have access to in Adobe Reader from JavaScript will be quite limited...