One of the easiest ways to fill door gaps is to use weatherstripping. You can use sticky foam tape; tubular silicone, vinyl or rubber; corner pads; felt rolls or one of the weatherstripping kits sold in stores. If old weatherstripping is already on the door, take it off and clean the frame with rubbing alcohol.
How do you close the bottom of a door with a gap?
If you have a large gap beneath the door, use a wrap-around door sweep. They have long adjustable sides that can be moved up and down along the width of the door. When the bottom of the sweep fills in the door gap, drill pilot holes through the sides and screw it down.