I recently got a new set of tires for my truck, which are not as heavy duty as the originals since I do not haul that heavy of that weight anymore. Do I follow the psi on the tires or what is on the door?
ALWAYS use the rating on door. The number on the tire it the rated BURST pressure. You should NEVER inflate a tire to that pressure cold. You would be BEGGING for a blowout when the tires heat up while driving. If the max pressure on the tire is not at least 5 PSI over the number on the door, the tires are NOT suitable for the vehicle.
Follow the recommended PSI specified on your door. The max PSI listed on any given tire is just that, a max. And should only be considered if and when hauling very heavy loads. If you, as you say, are not hauling any heavy weight, it is advisable to stick very close to the recommended PSI.
The rating on a tire is the maximum for that tire. As you say you have tires with a lower load rating then the originals, you will have to go by the rating on the tire. Additionally, the pressures on the door are based on the load capacity of the vehicle, not the tire, though obviously the tire also has to be able to support the load. That is why some cars have different pressures for the front and rear. A car, or truck, may not need to have the tires inflated to maximum, as the maximum load for the vehicle may be less then what the tires are capable of supporting. That is part of what was at issue with the Explorer/Firestone problem a while back. Ford recommended a lower pressure then the maximum for the tires. The only way to tell if your under or over inflating your tires is to watch how they wear. If the wear is in the middle, they're over inflated. If the wear is on the edges, they're under inflated.
