The title pretty much says it all. The installation process to get a fully working Java, Android and GWT development environment up and running can be a bit tricky at times. These three videos walk through the entire process, from downloading and installing a Java JDK, then the Android SDK and GWT then finally creating your first LibGDX project. The second video then looks at working with IntelliJ IDEA with LIbGDX, the third video shows the same thing for Eclipse. Each of these videos walks through loading a project, then running a Desktop, Android then HTML5 project. Each of the videos is hosted on YouTube and is available in 1080p. Click the link above the video below to open it directly in YouTube.
Part 2: Using IntelliJ IDEA with LibGDX
Part 3: Using Eclipse with LibGDX