How can I make the inside of a garage door not look so uggh?

Sounds like it needs a good cleaning - "always looks dirty & greasy", " huge grimy beast." Then I guess you could paint the inside (door, beams, hinges, and all) any color that pleases your eye.

Could you paint it a brighter lighter color....metal and all? Use a rustoleum paint. They are great for metals.

What about hanging interior curtains across the garage door opening? The curtains would have to be drawn back though every time you exit the garage, though, and you wouldn't be able to open the garage door from the outside without the door getting mixed up with the hanging curtains. If your door is uninsulated, you could try cutting and inserting into the inside of the door panels something like foamboard or posterboard cut to size and painted, wallpapered, or decorated in some other finish - similar to how you would insert insulation. You will still see the metal grid/frame, though, just not the inside of the bare metal door panels themselves.

You can fit foam panels into each section of the door. They add insulation to the garage and will give a clean surface to look at.