Pressing Tab in windows and dialogs normally just moves the focus between text boxes and lists. To make the focus move to all elements of the window, start by opening the Keyboard and Mouse Preferences (in System Preferences) and going to Keyboard Shortcuts.

Under the "Full Keyboard Access" section at the bottom, choosing "All controls" will allow you to focus on things like pop-up menus and buttons too.

Now, when a dialog box opens up, there will be a faint halo around one of the buttons, which you can move by pressing the tab key. Pressing the space bar will select whatever has the focus. The up and down arrow keys now navigate menus, and right and left arrows for folders and subfolders.