Dirt is dry mud. You find it on the ground where there are trees and plants around.
Dust is small particles that float in the air. When the sun shines through the window you can see dust in the air very well. It settles in rooms and to remove it you have to use a vacuum cleaner.
Grime is a slightly wet, slimy kind of dirt. It usually forms on things like taps, toothbrushes, ovens etc. You clean it off with an antibacterial spray.