How to switch between buttons (like OK and Cancel) in a dialogue box?

At the bottom of the "Keyboard Shortcuts" tab of "Keyboard & Mouse" system preference, click "All controls"
Dialog boxes will now have one solid blue selection, which is activated by the Enter or Return key, and one selection with a blue outline, which is activated by the Space bar. Tab or Shift-Tab will move the blue outline to the right or left.

I've been tabbing through Mac dialogs forever, using that All Controls switch. Not sure why it isn't working.

Now, if it's really just OK and Cancel buttons, you don't need to know anything other than the common shortcuts for those buttons. Typically, you can press Return or Enter for OK, and Cancel or Command-period for the Cancel button. If there is another button, you can often press the first letter as long as it is unique; for a Don't Save button, you can often press the D key. In less enlightened apps, you need to press Command-D for Don't Save.

Kappy's list of shortcuts above doesn't include any shortcuts inside dialogs, so it isn't going to be very useful.

Apps using the standard OS X dialog code respect the tabbing preference.

But some teams code with their own custom dialog code that does not respond to the tabbing pref. You found one of them, Final Cut, which as you probably noticed has dialogs that don't look like other OS X dialogs. Same with Photoshop and some other Adobe apps; Adobe seems to have their own dialog presentation code. Except for Lightroom, which does work with the tabbing pref.

It is a little odd that one of the apps that doesn't respect Apple's dialog settings is one of Apple's flagship apps.

Here's the list of shortcuts. If you open the Universal Access preference pane you can control which keyboard, mouse, etc. functions are changed.