Update for Docker versions that come with Docker.app
The Docker experience on macOS has improved since this answer was posted:

The only prerequisite is now for Docker.app to be running. Note that starting it on demand takes a while, because the underlying Linux VM must be started.

Any shell then has access to Docker functionality.

By default, Docker.app is launched at login time (you can change that via its preferences).
If you instead prefer starting and stopping Docker on demand from the command line, here are bash scripts that do that, docker-start and docker-stop; place them anywhere in your $PATH.

When docker-start launches Docker.app, it waits until Docker has finished starting up and is ready.