Here you can find a series of video tutorials on libgdx. I'd suggest watching them in order. The developed source code will be available in SVN shortly.

warning: these videos are old and out of date.

A Gentle Introduction to Libgdx

Learn how to setup a libgdx project for the desktop and Android.

Libgdx Tutorial #1 - Application, ApplicationListener and Life-Cycle

Learn about the basic application life-cycle of a libgdx application.

Libgdx Tutorial #2 - Basic File I/O

Learn about basic file input and output, using the abstraction provided by libgdx.

Libgdx Tutorial #3 - Basic Audio

Learn how to playback music and sound effects with libgdx.

Libgdx Remote

Fun with Gdx Remote (search on the Android Market for a compiled binary).

Android Game Development Series

Series on creating android games using libgdx.