When I was first started teaching myself programming, after finishing a tutorial I would feel like I still couldn't do anything in the language. So, I looked around to find something to work on. Since I had just learned a few of the basics, the amount of work involved in finding, reading and adding to an open source project seemed insurmountable. Instead I started on a couple toy projects, which ended up being incredibly beneficial.
Having seen a lot of questions from beginners similar to "what should I do now?" and a lot of answers similar to "start working for an open source project" has made me think there has to be better advice for a new programmer. While working on an open source project surely gives great experience, there is a perceptible barrier to entry.
Instead, I think it would be great if new programmers were prodded towards working on a toy program related to some interest they have. Since there are so many directions that programming can take you, I think it would be interesting to list some simple (but fun/rewarding) projects grouped by the direction the new programmer is looking to pursue. Such as:
Game Design:
Write a text adventure (like Zork)
Natural Language Processing:
Create a program that writes meaningless, but grammatically valid essays.