Start with fresh mushrooms
Wrapping them in a paper towel and then storing them in an open plastic bag (to prevent condensation from building up) might lengthen their shelf-life another day or so but can also darken the surface of the mushrooms.
When to wipe
Especially if you'll be eating the mushrooms raw (and they are not very dirty), it’s best to simply wipe them clean with a slightly damp towel or a soft brush. Rinsing them with water, on the other hand, can leave the surface of the mushroom slightly slimy—not a texture you’d want in a raw mushroom.
If necessary, clean with water
It’s amazing what will emerge from the little crevices of a morel when you wash it. Small bits of twig, hidden soil, and the not-so-occasional live inhabitant. I know it looks like the mushrooms are soaking here, but I had to pause the rinse for a second to admire the white maggoty thing coming out of the large morel.
Generally, cool water does the job. If you suspect there are things living in the mushrooms, warm water helps draw them out. Some people like to use salted water; I would follow up with a final rinse of unsalted water if using this method.
Dry mushrooms well and use soon thereafter
Trim the stem
Most other mushrooms, including white button, crimini, morels, portobellos, can be trimmed close to the base of the stem.
If you are stuffing the mushrooms, and therefore need to remove the stem entirely, reserve the stem for stock or chop it up and add it to the stuffing.
Removing gills and peeling are optional
Mushrooms don’t need to be peeled except for cosmetic purposes.
Trim away damaged spots and split up clusters
Mushrooms that grow in large clusters should be broken down by gently splitting them open along the grains of their growth.
Now you're ready to prepare the mushrooms whatever way your recipe or whimsy dictates. Keep in mind that, in general, wild mushrooms should be cooked before eating.