Brake lights and signal lights don't work

Those lights are on different circuits. The high and low beam lights are powered thru the headlight switch, the turn signals, brake, and hazards go thru the turn signal switch.
I would want to know if each circuit has power at the switch that controls it.

Do you have power going into the turn signal switch from the brake switch, and then coming out to go to the lights ?

I can almost 100% guaranty the turn signal switch is the problem. Power for the brake lights starts at the brake switch. power then splits from there and the third brake light gets power directly from the brake switch. Since the turn signal and the brake lights are on the same bulb power for the brake lights is routed through the turn signal switch. the switch depending on what position it is in will allow brake power or turn signal power to there respective left or right brake/turn bulbs.

If you hold your foot on the brake pedal and wiggle the turn signal and or play with the hazard switch the brake lights will almost always intermittently start to work. this will confirm a bad turn signal switch which is pretty common on GM vehicle with this style setup.

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The turn signals and brake lights use the same circuit and filament in the bulbs. The tail lights are on the other circuit which comes from the headlight switch.
Do you have power on the tail light fuse ?

sounds like you are getting water into the connections and possibly have a bad earth connection. Suggest you check all light connections for water ingression.

You may have a bad signal switch..Your brake lights run through the signal switch..Try hitting the brakes and move the switch around and see if your brake lights come on intermittently..

I suspect there is a problem with the turn signal/ hazard switch. turn the turn signal on left or right, if it doesn't flash, grip the hazard button and gently wiggle it around, if the signals start flashing, replace the turn signal switch. The Brake lights are wired through the turn signal switch, if the 3rd brake light works, the brake switch is OK. I don't think it has anything to do with the anti theft system (at this time)

need to check the multi-function switch. the brake light signal goes through the turn signal switch on the steering column. the brake lights come on if the turn signal is not being operated. the circuit for the brake light can loose contact in the switch.

Sounds like you have a bad multiple function turn signal switch. A couple things to try With your foot pressing on the brake pedal, Press the four way flasher button up and down, press and hold it down, pull up on it, or wiggle it. If the brake lights come on during any one of those replace the switch. With the key in the run position pull, push, the turn signal lever slightly. If the turn signal or brake lights start to work replace the switch. Let me know if none of those work, they usually do, and I'll get you some more in depth diagnostic info.

I’m happy to assist further over the phone at https://www.6ya.com/expert/jeremy_d728a59f986299fa

Replace the turn signal switch. All of these lights run threw it.