The Chef Wiki admittedly needs to be updated. We (opscode) have been working on getting the next big release ready, and with that will come a heap of documentation updates. Stay tuned to the Opscode Twitter Feed, Opscode's Blog, and the Chef Mailing list for news about the upcoming changes.
The Quick Start guide on the wiki is the simplest, easiest way to get started using Chef with a Server, by utilizing Opscode's Platform as the Chef Server. Once you get the hang of the workflow, you can set up your own server if you like.
The Chef Wiki has a page with Presentations that people in the community have given at various user group meetings and conferences.
Warwick's recent blog post is a good end-to-end overview/tutorial.
This is probably the video you saw, which is more of a sneak peak than an educational screencast.
Robert Berger of runa.com posted an extensive blog post about getting the latest version from the source up and running.
Ezra Zygmuntowicz of Engine Yard posted a "Chef 101" getting started tutorial shortly after Chef was released. It references an old version of Chef, but it covers many of the basics.
If you have questions, the IRC channel and Mailing list are great places to ask, and on twitter we use the hashtag #opschef.