The easiest way to see the Grand Canyon is to visit one or more of the many view points. In the summer with the throngs of visitors jamming the roads, the easiest way to get to the view points is by shuttle bus. The buses are free and no ticket is required. Stay as long as you like at places like Desert View on the east end to Hopi Point on the west---and then hop back on the shuttle bus. Stop by the visitors center at Mather Point to see displays and take in a ranger talk to learn about the geology and history of the Canyon. Check out the free newspaper you'll get as you enter the park for other ranger-led activities.
Walk it or Hike it
Take a walk along the Rim Trail, which is flat and paved. Walk a mile or two and then hop on the shuttle bus to another section. Get up at dawn or stay out until sunset and walk out to one of the vista spots like Maricopa Point to watch the Canyon turn deep red. Bring your camera and plenty of digital card space and/or film.
If you have some hiking experience, go down part of the way on the Bright Angel Trail or the South Kaibab Trail. They are both well-trod---and extremely steep coming back up. It's a difficult hike especially because you'll be climbing at a 7,000-foot elevation.
If you're in excellent shape, the Hermit Trail and Grandview Trail are more challenging, and if you're super fit, you could go all the way down to Phantom Ranch at the bottom of the Canyon. A back-country permit is required for overnight backpacking, except for guests staying at Phantom Ranch---which requires its own reservations (and you'll need them months in advance).
WARNING: The heat at the Grand Canyon is brutal in the summer.
If you like camping and welcome the thrill of river rapids (and don't mind the challenge of using a primitive outdoor toilet), take a river rafting trip down the Colorado River that cuts through the center of the Canyon. Commercial trips range from a few days to nearly two weeks. Boats come in the motorized variety or oar-powered. Book a commercial trip from an outfitter if you know nothing about rafting---otherwise, permits are issued a year in advance on a first-come first-served basis.
Take a helicopter ride to get a bird's-eye view of the Canyon. Rides range from 25 to 50 minutes on average and leave from the South Rim or from as far away as Las Vegas.
Go for a mule ride down to the bottom to save wear and tear on your knees. Book a year in advance, the rides sell out early. Opt for a one-day trip to Plateau Point down the Bright Angel Trail or stay overnight at Phantom Ranch at the bottom.
Hop aboard the old-fashioned steam train in Williams to save on driving. Trains depart from this little town an hour south of the Canyon mid-morning and bring you back mid-afternoon---or you can spend the night at a Canyon hotel.
Go in the Winter
Beat the heat, beat the crowds, go in the winter. The hotels are open, the shuttle buses are running and if there is snow, the staff will plow the roads. Bring crampons and hiking poles if you want to hike the Bright Angel or Kaibab in the snow and ice---they can be treacherous. Otherwise, bundle up and enjoy the views.
The North Rim
The North Rim is 1,000 feet higher and hence cooler than the South. It's also more remote, so fewer people visit it. You can still hike down into the Canyon via the North Kaibab Trail (not in the heat, not for the faint-of-heart), and you can also do the easier interior hikes, like the Widforss Trail and the Cape Royal Trail. Go for a mule trip---a one-hour on the rim, a half-day on the rim or a half-day down in the inner canyon (they don't go all the way to the bottom).