When I started learning Django, the first thing I did was work through the tutorial in the official documentation.
I cannot stress enough how useful the official documentation is. Not only is it chock full of useful code examples, but the fact that it is a living document means that it is always up-to-date (unlike Django books or blog posts).
I can not really think of any useful Django applications on Github or Bitbucket that are simple enough for someone who is learning the framework to replicate, but if pushed I would suggest:
- the source code for the framework itself
- applications made by core developers (like django-registration or django-taggit)
- applications with a significant following in the community (like south or django-sentry)
