We've gotten to the point where we always run pretty much the same pressures since we run MG Yellows/Evincos all the time.
But what I like to do starting on a new tire, is set it at the manufacturer's specification first. Most brands have a recommended race pressure for each compound. From there, keep dropping pressure until you feel the sidewall begin to flex and fold over. That's your minimum pressure. To get your maximum pressure, keep going up from that recommended psi until you hit a point where the kart starts to become super loose (tires overheating) after 2 laps. That's your qualifying pressure.
If it's colder and lower grip, run closer to your qualifying pressure. If it's hotter and high grip, run closer to your minimum pressure.
We don't do a lot as far as staggering pressures, but sometimes if the track is really left or right turn biased we will drop .5 on one side and go from there.
Once you get a feeling for the tires at their lowest and highest workable pressures, you can start to feel how different pressures actually affect the balance of the kart. I've always found on a soft tire like MG Yellows, the lower you go the more dig you get in the corner. If you go too lower though, the tire will start to fold over and hop.
Example: MG Yellows run best between 8-10 psi. If you go lower, you risk folding the tire over during hard cornering as the sidewall gives way. We've gone to 7.5 when it's really hot out, but it's rare. At 10 or higher, you're starting to overheat the tire really quickly. With higher pressures, the tire starts to bulge and bit and doesn't bite into the track as much. It skates around on top of the rubber rather than digging into the rubber.
We did the LeCont tire test last year and used this method and were able to dial a completely unknown tire in, in just a few sessions.