Before
Before you put your first crease in a piece of paper, pick it up, and familiarize yourself with its feel. Is it stiff or flexible? Is it smooth or rough? Are the edges cut cleanly or does it have a ragged edge? Does the paper bend nicely or does it feel like it is brittle like a dry cracker as if it were going to break as you bend it? Good folding paper should be thin and lightweight so it does not get too thick with many folds. It should be strong and withstand at least a few folds back and forth along the same line.
During
Feel how the paper handles under your fingers. Crinkle it gently and feel its resistance. Use pads of your fingertips: they are very sensitive and can detect tiny differences in surface and edge features. In fact, when folding a square in half, if you adjust the paper so the edges touch your fingertips you can tell when the edges are lined up exactly even without looking: your fingertips will tell you. Then grasp both layers, don't let them slip, and make the crease without letting go. Same for a diagonal: as you position one corner over the other, let your fingers feel when both sides of the corner are aligned, then grasp the layers together and make the crease.
When forming a crease, start with a wide curve and loosely holding the paper. Gradually (don't rush!) make the curve smaller and tighter, feeling the curve form itself into the crease. When the curve is small and tight, go ahead and press the crease into the paper.
After
Forming the crease in a high quality origami paper can be gentle (using the pads of your fingertips) or firm to very firm (using a fingernail or the slick side of a sharpie pen) to press the crease into a permanent mark. In most standard pieces, a good firm crease can be obtained with a fingernail or hard object making the two layers look and feel like one when you are done.
Look at the finished crease after you are done: are the edges lined up and straight? Did anything shift between when you knew the edges were lined up and after the crease was completed? Is the diagonal fold on a square coming to a perfectly symmetrical point (if it's off center one side will show more than the other)? Did a part of the paper get stretched from too much pulling or pushing? Imperfections are just opportunities to practice and make it better, but you have to look carefully at your work to verify that your crease is as good as it can be. Don't just fold and look away.
Practice
Becoming familiar with your paper and how to handle it, how to make it obey your objective, An exercise that can be done anywhere, is to shape an edge of a piece of paper into an "S". Start with wide soft curves, but tighten them bit by bit, and as the curves get tighter and smaller, line up an edge with the crease. Adjust the lengths of each of the 1/3 so they balance out and converge on an exact 1/3 of the edge. When the curve gets small enough right when the lengths are visually equal, set the crease firmly.
Creasing Many Layers
When taking a piece and making a crease when there are already many layers in place, it is important to fold all of the layers together, and very important to prevent the inner layers from slipping. One way is to grasp all the layers firmly on both sides of the crease while the fold is being formed. Don't let the inner layers slip away from the crease: this can easily happen if there are many layers already in the piece being folded.
Fold on!
