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.

The others have given good descriptions. To me, a more specific difference is that 'grime' is 'dirt on something that should be clean'. As Ariel said, it can be on household surfaces. Another common use of 'grime' is to refer to the dark stain on the inside of a shirt collar, but it could also refer to any dirty mark on your clothes.

Practically speaking, dirt and grime are used interchangeably (the same). Dust is what forms in the air from dead skin and other things. Run your finger on a shelf you haven’t touched, or a book you haven’t read in a long time. Is there something on your finger? If so, that is dust.