No matter how good your lace looks when you first tie it off, it will look better after you press it. If a piece is made in rounds, it is a good idea to press after every tying-off, so that you can see how you are progressing. If you are making up a pattern as you go along, pressing after every round will be necessary to let you know whether or not you are keeping flat.
To press tatting, put it face down on a padded surface, cover it with a damp cloth, and press straight down with a hot iron, then lift straight up. Uncover the lace and leave it where it is to cool and dry, or carefully move it to a towel-covered surface. (If you habitually press more motifs than you can leave on the ironing board, a thin, stainless, restaurant-size pancake turner can be useful. )
You may count this steaming as a wetting and drying for the purpose of securing your thread ends, but it is better if you put a drop of water on the knot itself before you plug the iron in to heat it. If the motif is intended to be washed, particularly if it is to be appliquéd to cloth, a thorough wetting now can save surprises later. It will take at least ten minutes of submersion for water to penetrate the knots. Do not attempt to iron soaked lace completely dry, just steam it thoroughly (using a dry cloth to protect the lace and absorb water) and then leave it to air dry. A second pressing when it is nearly dry may be in order.
If you use two colors of cheap thread to make something that isn't intended to be washed, be careful not to get it wet during pressing, because the dye might run. Such work is impossible to stiffen, but tatting usually has enough body by itself, so you can use bleeding colors for quickie work. If you plan something elaborate, don't take the manufacturer's word for the quality of his dye; make something small, sew it to something white, and wash it. Repeat with every dye lot -- everybody goofs sooner or later.
Stretching all the picots while the lace is damp improves their appearance, and causes the lace to dispose itself more symmetrically. A crochet hook is the handiest tool for this, because it is stiff and yet tapers to a fine point. A knitting needle fine enough to enter the picot might be uncomfortably flexible.
There are discussions of "Washing Tatting" and "Stiffening Lace" in the Encyclopedic Index.