The world of AWS is huge these days. The HashiCorp tools can be counted two hands, but since they interface with AWS, that limited count is deceiving.
I actually started by getting the lay of the land from AWS in Action, which Manning conveniently had on special just about the time I was interested in learning more about it. The book isn't an exhaustive coverage of all of the AWS services, but it's an excellent overview. I did their tutorial for building a WordPress site. The authors provide their code examples online in GitHub, which is excellent.
I don't for a minute think that doing either of these tutorials qualifies as any real expertise, but I found that just typing the command and checking the results in the AWS console starts to build both the physical and mental "muscle memory." Both tutorials were done with the free tier in AWS, and free (as in beer) is always good.
Next, I checked out a new tool from HashiCorp - Otto. Otto is a successor to Vagrant, but it's heading in very different directions. If you start thinking that Otto is just Vagrant++, it's hard to understand the infrastructure and deployment functionality that Otto provides. Otto provides a path from a development environment on a single machine, to a simple AWS deployment, to a more sophisticated AWS deployment.
Because Otto is based on some opinionated policies and best practices, it provides a great way to see how all the pieces of the HashiCorp ecosystem and AWS fit together. It generates plain-text configurations and scripts in the .otto directory in your project's tree. These are there to read and learn from. Some AWS masters might chafe at the best practices, but everyone's gotta start somewhere, so it might as well be something sane.
I'm not sure if this is the best way to learn about AWS and the HashiCorp tools, but it's what I've done. Your milage may vary.