Turn your furnace off, or set the temperature low enough that you're sure it won't kick on and blow air on you while you're working.
Put on your dust mask. Starting at the highest heating vent in your house, remove the vent by unscrewing the mounting screws. Set the vent on an old towel. Run your vacuum hose over it to pull out dirt and hair in the grating. Put on your rubber gloves, and use your bucket of soapy water and a sponge to clean the surface of it.
Put the extension hose on your vacuum, and begin vacuuming down into the duct. Make sure to get any dust and dirt you and see clinging to the sides. Reach down as far as the hose will go.
Wipe the inside of the duct with your damp sponge, reaching down as far as you can and getting all the sides. Pat the grill dry with your towels and put it back on with the same screws.
Repeat the process at each vent in your house. At the end, put your new furnace filter in, then turn the furnace back on.