Holy smokes, it's amazing that a few dudes chiseled all those illustrious (and enormous) presidential faces, and if you haven't already paid tribute to their artistic skill, it's not too late. Just don't go on a stormy day like I did, when everything but Washington's nose was obscured by fog... It was a long drive just to see a nose.
Almost 3 million people visit every year, with the most hectic months being June, July and August. May, September and October are good months, as they're less busy and not so hot. But if you do go in the fall... don't go during a whiteout.
While there aren't official fees for the monument (you can drive through and see the monument from your car window or at pull-outs), there is a parking fee for the main parking lot, which can be found here. It's a little lame to drive all the way there and not park, but just beware long lines and traffic jams.
Pets aren't allowed into the main viewing area, and are relegated to the parking lot and surrounds, as outlined on this map.