The J Studio
Help|Studio from the J menu puts you in the J studio, where you can run labs and demos.
The labs are interactive tutorials. Some describe the J language, some describe addons, others describe J applications. They are all divided into short sections, each of which executes a bit of J code and discusses it. To begin with, while you are just marvelling at the terseness of the language, you should just go through the lab with your eyes open, seeing but not fully understanding.
When you run a lab, press Ctrl+j (Command+j on a Mac) to move to the next section.
Demos, launched by Help|Studio|Showcase or Help|Studio|Qt Demos, are programs that do something cool. In many cases you can see the demo code and its compactness.
A Beginner's Path
Here is a good sequence for the new user to J. See the elegance and speed of the solutions, without trying to understand the details.
Viewmat is a quick visualizer for arrays. J uses lots of arrays. But then, all languages use lots of arrays. Why don't they all have viewmat?
J's plotting package is comparable to any commercial package, plus you can call it from J directly. This makes plotting especially easy.
Run Help|Studio|Showcase|Plot and select different plots. For ones that strike your fancy, click Options|View definition to see the code behind the plot. Isn't it tidy?
For a walkthrough of plotting, run Help|Studio|Labs|Plot Package.