How do you change the bulb on the fog lamp for a 2003 Pontiac Bonneville?

Many things could be your problem.
1- light bulbs (most common)
2- no ground or B+
3- defect fog lights switch
You have to find out with a test light or meter.
At the fog light connector. See if you have B+ and good ground.
If yes then the bulb is the defect.

Instructions on how to change a headlamp bulb are on page 5-56 in the 2003 Pontiac Bonneville Owner's Manual.
If you do not have an Owner's Manual, you can download a PDF version at the link below (direct link)
2003 Pontiac Owner's Manual

You can check for power where it goes into the fog light switch and then the output terminal when the switch is on, or simply skip that step and turn the switch on and test for power at the fog lamp connectors if you have no power go back and check the switch if you do have power make sure the fog lamp housings are properly grounded and then test with a new bulb

on the back side of the light turn it counter clockwise bulb will come out unplug and reverse the steps

First remove the driver's side fog lamp plastic cover out from front bumper. After that you will find torx socket head panel screws from fog lamp. See picture below:

No fuse for lights on vehicles for the element in the bulb acts like a fuse. They all have relays only

remove lower cover, remove cover on back of lamp and disconnect wire then unhook clip (clip just swings to the side) and remove bulb. dont touch new bulb on the glass part ,will weakin life of lamp. goodluck.

Could be both the bulbs are burnt out so replace the bulbs if bulbs are good check fuses if fuses are good the check the wiring.

check ground at fog lamps.

1. Using a thin bladed tool, carefully pry the lamp lens from the overhead console.
2. Using a thin bladed tool, carefully pry the metal shield inward until the retaining tabs clear the notches.
3. Remove the metal shield.
4. Remove the courtesy lamp bulb.