The palms of my hands really hurt I understand that the pressure on my hands causes it.
I also understand how the nerves run through my hand.

What I don't understand is how to implement the advise of, "Take the weight off your hands"

A fellow told me, "Put 90% of your weight on the pedals, 10% on the seat, and barely touch the bars with your hands." I dunno how to do this!

With the seat up in order to get a full pedal, the angle of your body is going to naturally push down on your hands. The only way I know to "put weight in the pedals" is to stand up.

Short of getting a raised handle bar or lowering my seat, how do you get the weight off your hands? Is there some kind of posture magic I can't figure out?
You are correct that a leant-forward riding position will naturally tend to make a rider support some of their weight with their arms and hands. The more rotated forward the rider is, the more weigh the arms tend to have to support - which is one reason tri-bars have forearm rests.

Good core strength is what allows a rider to take weight off the arms. Strong lower back and abdominal muscles support the torso, taking weight off of the arms. You can try doing some core strength building exercises (which will likely make you a stronger rider and help prevent injury as well). 

You should look at your fit on the bike, you may in fact have the bars too far forward and down. Raising and pulling them back may help while you build core strength.