The inside of my car sticker tells me that at cold temperature, the tires need to be around 33 psi. The tires themselves say 51 psi. Which number should I go by?

 

If,,,,,,,,,,,,,,,,,,
you are running the size/load range/speed rating tires the car was certified for, you should use the pressures on the certification label sticker affixed on the vehicle.
A lot of folks are not aware when you alter those certifications some insurance companies can walk right out from under any obligation in the event of a mishap.

 

Also, tire pressure increases as tire temperature increases. On my trip to AZ in my F-150, mu tire pressure monitors showed 38 PSI cold on all 4 tires before the trip and was running 43-45 PSI from Dallas, TX to Queen Creek, AZ. just last month. Tire pressure increases and decreases 1 PSI per change in air temperature.