How do I maximize the Safari window all the way?
The green button is in fact the Zoom button, and does not serve the same purpose as a Maximize button in for example Windows would. You can read more about the differences between Mac OS X window controls and Windows window controls here: http://www.davidalison.com/2010/11/window-controls-mac-os-vs-windows.html

In short, whereas a Maximize button's purpose is enlarging the window to take up the entire screen estate, the purpose of a Zoom button is to enlarge the Window based on the resolution and the user interface inside the window.

Some applications might implement an Option-click for a maximizing behavior, instead of zooming, but this is application specific and Safari does not have such a functionality. Instead, Safari will "zoom" the window based on the content of the window, which usually just means enlarging it vertically. If there are horizontal scrollbars, Safari might also enlarge the window horizontally to try and eliminate the scrolling.

As an alternative, you can use fullscreen mode in Safari which will always take up all available desktop space.

If that doesn't suffice, there are some additional window management tools you can install which facilitate window placement and scaling. I can recommend Moom, which is available in the App Store.