I'm looking specifically for shortcuts that resize, move and otherwise let me avoid using the mouse for manipulating application windows. Bonus points for shortcuts that let me move windows between different monitors.
Although not "offical" shortcuts Alt + Space accesses the system menu so:
Not directly shortcuts, but I use Enso to:
Among other things.
What I love from this is exactly what you're asking. Not having to take my hands out of the keyboard to do this.
It doesn't support "move to monitor" though