I have a 2002 Silverado LS, and I have never really had any problems out of it. About two days ago I got pulled over because my brake lights weren't working. Anyway, after tinkering around trying to figure out what was wrong, I realized that everything works perfectly only when the headlights are on.
The truck has automatic headlights and daytime running lights. Well, the brake lights only work if I manually turn on the headlights. I also noticed that the daytime running lights aren't coming on at all. I really think it has to be a switch, fuse, relay, etc (at least I hope it is), but I just don't know. The Chevy dealer told me it would cost $110 just to look at it. I just don't want to pay these guys that much to replace a fuse.
Late model exterior lighting circuits are not as simple as way back when, and your brake lamp problem can have a number of potential causes. But you're right, start with the basics. Check all the fuses (under hood and under dash), especially the 20 amp "Stop LPS" fuse located in the under hood fuse block.
The 12volts for the brake lights originates at that fuse (assuming there is voltage at the fuse block feeding the fuse). From that point, power goes to the brake light switch which is located at the brake pedal, and then past a few other connections before actually reaching the brake light bulbs. The brake lights circuit is "hot at all times" and should work even with the headlamp switch and the ignition switch in the off position.
Just a guess, but it sounds like that fuse may be burned out and you're getting feedback voltage from another circuit when you turn on the headlamps. Of course, the cause of an overloaded circuit which burned the fuse is another ball of wax.
If all fuses are good, and power is applied to the brake lamp switch with the ignition and headlamps off, it gets more complicated. If that's the case, it may be worth the diagnostic charge at Chevy service to save you the aggravation.