Maps in the literature given out at the entrance stations show parking areas in Grand Canyon Village. Once you've found a spot, the free Village Shuttle will take you around.
Cycling is another option, though it's restricted to roads and bike paths.
Walking, especially on the very scenic Rim Trail, is the most enjoyable way of seeing the sights here. Eventually parking will be moved outside the park to Tusayan, where buses or a light-rail will convey visitors to Canyon View Information Plaza and beyond. This new visitor center complex, a short stroll south of Mather Point, opened in late 2000 and has replaced the old visitor center. Because it's designed as part of the public transport system, there's no parking here. You can take the Village Shuttle or walk via the Rim Trail.