Intro

You can download Blender from blender.org.

If you have a Mac with Homebrew, install Blender from the command line:

$ brew tap caskroom/cask
$ brew cask install blender

Blender has a reference manual, but it is easier to learn from YouTube videos. I started with this series of videos, which runs about 3 hours:

Blender Absolute Beginner Tutorial: Episode 1, Ep. 2, Ep. 3, Ep. 4, Ep. 5, Ep. 6, Ep. 7, Ep. 8, Ep. 9, Ep. 10

The tutorial shows how to:

change your perspective in the 3D View
select objects
create primitive shapes
move shapes in the 3D View
use modifiers, which act on shapes to produce new shapes
use textures, which govern the look of the surface of shape
render a scene and save the render as an image in a file

The tutorial ends with a physics simulation which is an advanced topic but it gives a sense of what Blender is capable of.

After you have the basic skills above, you can often make progress on projects by doing a Google search for "blender tutorial how to X" whenever you get stuck.

Blender provides a tiling window system. This video shows how to manage windows:

Ira Krakow's Blender 2.5 Splitting / Joining Windows Tutorial

The tutorial shows how to:

subdivide a window horizontally or vertically
resize a window
change the type of a window
merge two windows into one
show or hide panels in a window
use control up/down arrow to toggle full screen for a window

Blender windows have "headers", which by default are on the bottom of the window. The leftmost element in the window header is an icon which is actually a dropdown list. It can be used to change the window type. There are 17 window types, including these 5 which are worth discussing now:

Info
User Preferences
3D View
Outliner
Properties

Info Window

This window is often at the top of the screen and consists of a single header which serves as menu bar for Blender.
It contains the screen selector, the scene selector, and the render engine selector.

The screen selector has several screens, each with predefined choices for windows and their types. The windows in each screen can be customized, but if you save the blender project and open it again with New the original screens will be restored.

User Preferences Window

Also available from File | User Preferences....

Blender assumes the presence of a 3 button mouse and a numpad. If you don't have them, as is often the case when working on a laptop, under the Input tab there are checkboxes for "Emulate 3 Button Mouse with Alt+Left Mouse" and "Emulate Numpad".