I was browsing around the Wikipedia articles for Cellular Automata recently, and the thing that struck me about it is how easy it is to notice, if not understand, the levels of abstraction. You start with the rules of Conway's Game of Life and a few samples, easy enough. You think in terms of what a cell does. Then you learn about oscillators, periodic self-sustaining patterns, which aren't that surprising, and still on the level of individual cells.

Then you learn about Gliders, a simple pattern that in four generations gives rise to a copy of itself, but translated 1 cell away. Now we're not talking about what cells are doing anymore. We're talking about what patterns are doing, and they can move around independently from the cells used to represent them. It's a layer of abstraction.

You find out about objects like Reflectors, where when a Glider collides with it, the result is a Glider in a different direction, and things like the Glider Gun that periodically produces a new glider. You start arranging these things together to get a nice little circuit, and soon the Gliders aren't even the object of study anymore, they're just little blips that other things use to pass information. Instead of just the patterns, you now think in terms of the interactions between the patterns, which is another layer of abstraction.

And you can grok these successive layers of abstraction in 15 minutes of reading a little explanatory text and watching animated .gifs. I think the fact that Conway's Life is inherently a very visual phenomenon makes it easier to understand like that.

The article covers a lot of breadth of cellular automata. I think you get a lot more out of it if you have a tiny bit of depth in one particular automata first, so you can understand in a general sense the cool patterns and phenonema that it talks about. Quick reading on Conway's Life (selected with order in mind):

http://en.wikipedia.org/wiki/Oscillator_(cellular_automaton)

http://en.wikipedia.org/wiki/Glider_(Conways_Life)

http://en.wikipedia.org/wiki/Gun_(cellular_automaton)

http://en.wikipedia.org/wiki/Rake_(cellular_automaton)

http://en.wikipedia.org/wiki/Breeder_(cellular_automaton)

http://en.wikipedia.org/wiki/Reflector_(cellular_automaton)

Be sure to click through on the preview images to the animated version, especially rakes, breeders, and reflectors.