<p>I downloaded Squeak, the Image file, and the Sources file, but I can't figure out how to write Smalltalk code. Is there something obvious I'm missing? Is Squeak even the right IDE for serious Smalltalk development?</p>

I don't think Squeak is really oriented towards "serious Smalltalk development". It's intended as a first programming environment for children. Which isn't to say that you can't do  useful stuff with it, it's just not aimed as much towards large-scale development.
Check out the
Wiki
for getting started tips. It's been a while since I last used Squeak, but I don't remember it being particularly hard to get started.