THE TWO-DAY descent into the Grand Canyon atop a sure-footed mule has become one of America's famed tourist institutions, annually luring thousands of the adventurous. Years ago, composer Ferde Grofe paid musical tribute to this popular trail ride in his "Grand Canyon Suite."

"Plop, plop-ploppity, plop," sing the hooves in Grofe's music, beating out a cadence that suggests the mules' relaxed, ambling gait as they pick their way cautiously down the steep, rocky Bright Angel trail.

"Thump, thump-thumpity, thump," pound the hearts of the riders, who cling nervously to saddlehorns while peering into the mile-deep chasm from their uncertain perch. Grofe may have ignored this aspect of the trip in his composition, but I suspect, after making the ride with my wife in July, few tourists fail to mention it in their postcards back home.

We were on a two-week, 1,900-mile driving tour of the Southwest, sampling the scenic and recreational pleasures of the breathtakingly beautiful canyon country of northern Arizona and southern Utah. Our mule trip was a planned highlight that exceeded expectations.

Anyone afraid of heights should skip the overnight mule ride, the national park concessionaire advises wisely. Over a length of nine miles, the trail plunges in seemingly endless cliff-hanging switchbacks until--about five and a half hours by mule later--it reaches the cool, swift-flowing waters of the Colorado River that over the centuries carved out this natural wonder.

With nerves under control, the view--and the experience--are spectacular. Standing on the canyon rim, you sense the vastness of this 277-mile-long, up to 18-miles-wide gash in the earth's surface. But you must descend to the bottom and look back up the smooth, sheer cliffs to appreciate how very deep a mile really is.

At every twist of the trail, the scene changes. A new gorge appears, a stream emerges from a spring and tumbles alongside, sunlight playing on the rocks creates a rainbow of reds, a scrub tree struggles to grow on a lonely ledge. In the descent from a 7,000-foot altitude to about 2,400 at the river, canyon walls reveal a geological record dating back hundreds of millions of years. At the same time, you are dropping from the coolness of a Ponderosa pine-covered mountain plateau to the dry, searing heat (100 degrees or more in July) of a desert where cactus thrive.

My mule, Betty, takes it all quite calmly, pausing sometimes to nibble at a tasty bush on the precipice and then plodding ahead to catch up with the rest of the bunch. The animals make the trip regularly throughout the year and, according to trip guides, have never lost a paying customer. This nonchalance goes a long way in easing a rider's fears, who, with the exception of one rest stop, must remain astride the mule until it reaches the canyon floor.

At the bottom is an oasis of leafy cottonwoods nourished by the Colorado and its tributary, Bright Angel Creek, where exhausted riders eagerly soak their dusty, overheated bodies. In the shade alongside the creek sits Phantom Ranch, providing a welcome beer on arrival and later a steak dinner, shower and overnight accommodations in rustic (though air-conditioned) cabins or dorm. Nearby, the park service maintains a campground for the numerous hikers carrying their own food and tents.

Night falls quickly in the canyon, where the quiet is broken only by the sound of the Colorado spilling over its rapids. The stars are brilliant against the black sky. In their cabins, the riders (17 of them on this trip, ranging from teen-agers to a grandmother in her sixties) treat the day's saddlesores and then crawl into their bunkbeds for dreams (or nightmares) of the trip back up. There are only two options, riding or hiking out--and for the long uphill trek, the mules are in better condition.

A clanging cookhouse bell at 6 a.m. announces breakfast, a hearty meal of pancakes, eggs and bacon. The idea is to get back on the trail before the day's heat has a chance to fry the brain. Ascending, the route is via South Kaibab trail, about two miles shorter but much steeper and more precipitous. Yesterday's views, and terrors, were but a prelude to today's awesome four and a half-hour climb.

Riders who the day before took in the vistas with little apprehension, now only glance out of the corner of their eyes, which they keep focused, instead, on the rump of the mule ahead. If you don't see the dropoff, you reason, it isn't there. The mules seem testier. They nip at the beast ahead or let loose an irritable kick at their colleague behind. Even at this early hour, the sun beats down, and there is no shade except the wide-brimmed hats that are required wearing.

Up we go, past Panorama Point (what an unforgettable view of the canyon and river in both directions), past "The Tipoff" (another vista point) and Skeleton Point (reached by a particularly horrendous series of switchbacks). The brave let loose of the saddlehorn to snap a picture or two. No one says much, except to urge a mule homeward.

Finally, the top is in sight ("Thankgod"), though as if to test us, the trail once more spirals dizzyingly on the brink before depositing us at last on safe, level ground--wind-burned, dust-blown and thirsty, but also deeply moved by what we have seen.

And maybe we are just a little proud at having survived the physical and mental rigors of this rather strenuous excursion--at least until a well-muscled, darkly-tanned hiker in cutoffs and headband emerges from the trail and scoffs at "the dudes" who, he jokes, ought to try it next time on their own two legs.

The Grand Canyon is the attraction that draws many, if not most, tourists to the Southwest. But even if the canyon did not exist, the region would still be richly endowed in sightseeing, hiking and riding possibilities. It goes by many other names:

Canyon Country, for the countless gorges, gullies and gulches that slash through the high Colorado Plateau encompassing the Four Corners states of Colorado, New Mexico, Arizona and Utah. In Utah's Zion National Park, a small gem in the park system, smooth-faced cliffs tower dramatically hundreds of feet above the narrow Virgin River valley.

Colorland or Red Rock Country, for the rich reds and pinks of the buttes and mesas.

Geological Disneyland, for the unusual rock formations that erosion has carved. Bryce Canyon National Park in Utah is a Fantasyland of red-rock spires and cascading stone. Horse trails and moderately strenuous hiking paths weave through "Queen's Garden" or under the arches and up the switchbacks of the "Peek-a-Boo Loop."

John Wayne Country, because of the many westerns, such as "Stagecoach" and "The Searchers," filmed with nature's monumental architecture as a backdrop. Out West is the only place you can wear a cowboy hat and boots and not look foolish.

Butch Cassidy Country, after the outlaw made famous by the Paul Newman-Robert Redford movie. Cassidy roamed this arid, hide-and-seek land of hidden caves and box canyons, and a young Zion park guide from a neighboring community claims he is a descendant through his mother's side of the family.

The Navajo Nation, a 16-million-acre chunk of the region that belongs to the rapidly growing Navajo population, where a visitor sometimes feels he or she has stepped back in time. One Sunday, at Canyon de Chelly National Monument on the reservation, we climb onto the back of a flatbed tour truck and splash up the river bed where crowds of Indian families are gathered for a summer outing. Admittance to the floor of the valley is prohibited to outsiders unless a guide accompanies them.

Along with the scenery and a visit to the ruins of ancient cliff dwellings, we are given a glimpse of these families at leisure. I don't blame them, though, for resenting our intrusion, for we must look odd bouncing in our truck on our waterway path past their picnic and swimming sites. A couple of vans speed by, splashing us. We interpret this, perhaps naively, as good-natured fun.

The Southwest, additionally, is a land of broad vistas, where the distant horizon is visible in every direction. Empty two-lane roads stretch ahead for miles, but the view is never dull, broken by mountain peaks rising above 10,000 feet, high mesas or a jumble of colorful rock formations. One day, under bright sun, we watch a pair of rainstorms chase across the range far off to the west.

The region is sparsely populated, and towns are few, mostly crossroads service stations. A cautious driver fills the tank before it registers half empty because the next fillup may be a half a tank away. In two weeks, the biggest town we stayed near was Page, Ariz., with about 5,000 residents. Much of the area is rangeland for cattle, sheep, goats and horses. We spot Navajo shepherds tending herds and flocks near their hogans.

Except for an occasional rain squall, the sky is bright and sunny, a deep blue in the clean, clear air. Days can be hot, but they are not unpleasant since the humidity is low (10 to 30 percent). Because of the high altitude, ranging from 5,000 to 10,000 feet, cool nights predominate, and several times we need jackets. What a relief from sweltering Washington.

Vegetation varies from desert sage at lower elevations to aspens, spruces and firs as the road climbs. On the north rim of the Grand Canyon, at 8,000 feet, we see patches of snow preserved in the dark shade of overhanging rocks. Because rain is infrequent, most of the streambeds (they call them "washes" here) are dry, but a sudden downpour can turn them into racing torrents. Beware of flash floods, warn countless roadside signs.

On the Utah border in Arizona, the massive Glen Canyon Dam rises hundreds of feet between red sandstone cliffs to harness the flow of the Colorado River. Built between 1956 and 1964 to generate electricity--over the strenuous objections of conservationists--the dam has formed a huge lake, Lake Powell, which stretches 180 miles upstream along the former Colorado River channel.

Though the lake flooded canyons of incomparable beauty, say the conservationists, it has become a popular site for water sports, especially powerboating. Boaters can explore these canyon waterways (as many as 90 of them) leading off the main channel, and many take tents to camp and swim in isolated coves reached only by boat. Others rent houseboats and anchor in a hidden twist of a tributary canyon.

To see what we can of the lake in a short time period, we take the seven-hour, 100-mile motor yacht cruise ($40.50 per person) departing daily from Wahweap Lodge, a modern motel overlooking Lake Powell at the National Park Service-administered Glen Canyon Dam and National Recreation Area.

At 26 miles per hour, we plow upriver through the emerald-green water between reddish cliffs growing increasingly higher. Our principal destination is Rainbow Bridge National Monument, a soaring red-rock arch as tall as the U.S. Capitol. Until the lake was created, Rainbow Bridge was reached chiefly by an overland hike through rugged backcountry. Now thousands visit by water every summer.

En route, our boat pokes into Cathedral and Cascade canyons, cruising slowly into the twisting passageways until the polished cliffs all but graze the sides of our craft. A couple of times we surprise families of campers, whose hideaways were not so remote as they had thought. When our boat can advance no farther, the guide puts it in reverse until he finds a place wide enough to turn it around.

Just below Glen Canyon dam, where the controlled flow of the Colorado is released to continue its journey to Lake Mead in Nevada, is the launch site for the famous eight-day (or more) rubber-raft expeditions through the Grand Canyon. We have no time for that trip, but in nearby Page, the John Wesley Powell Memorial Museum (he was the first to explore the Colorado River's gorges) offers daily one-day raft trips ($40 per person) from the base of the dam to Lee's Ferry, about 15 miles downstream.

While this stretch of the river is free of rapids, the rafter does get the experience of drifting between the canyon cliffs, a relaxing and scenic ride without the roller-coaster thrills of the longer trips. The river, like Lake Powell, is a gorgeous green, and the contrast with the red-hued rocks is a constant delight.

Down in the canyon, the sun burns hot, and several rafters jump into the fast-moving water to cool off. Since the river is frigid, they scramble out in a matter of moments. A brief, if somewhat incautious, water fight erupts between the three rafts, cooling the rest of us.

The sudden rain squall hits, stirring up clouds of red dust, just as we are to descend by four-wheel-drive van down a winding, rutted trail into Monument Valley, a region of giant red-rock formations rising abruptly from the desert floor. Our Navajo driver and guide brakes, and we wait until the brief storm has passed. He is, it turns out, checking out the very real possibility of a flash flood, which would make our passage impossible.

As a result of the Hollywood westerns, these massive sculpted rocks have become the classic representation of the Old West. In the 1930s, Harry Goulding, founder of Goulding's Lodge, where we are staying, helped convince director John Ford to film "Stagecoach" in the valley. Later, the lodge's dining room was built for the set of "She Wore a Yellow Ribbon," and photographs from this and other Ford films line the walls.

The valley, located on the Navajo reservation, is officially a Navajo Tribal Park. Several families still live among the stately buttes and pinnacles, some of them tending the flocks of sheep that graze among the yucca, sage and juniper. Our four-hour tour ($20 per person) provides close-up views of the rock formations as well as our only chance to step inside a hogan.

Built usually of logs, stone and mud, hogans traditionally are round with the door opening east to face the morning sun. We visit such a structure belonging to Susie, an elderly woman dressed in a blue velveteen blouse and long brown skirt. A large silver and turquoise necklace tumbles down her chest. Though there is a small house close by, Susie keeps her loom for weaving rugs in the hogan. For a small tip, she demonstrates the intricate, time-consuming process.

At another stop on our tour, a Navajo boy rides by on his pony, halting on the edge of a gully to stand quietly peering off into the distance. With the red-rock monuments in the background, it makes a classic photograph. So what if it's a gimmick--the lad returns a few minutes later for his tip--it delights the tourists.

The sense of remoteness a traveler from the crowded East experiences in the Southwest is easily illustrated. Few of our rooms--we stayed mostly in national park lodges--had radios or TV sets, and no place was large enough to support a daily paper.

On a Friday, while we are trail-riding at Zion Canyon, a major development having international implications is taking place in Washington, D.C. No one we talk to later in the day mentions it. So it isn't until late the following Tuesday, when we switch on the car radio during the long drive to the Grand Canyon, that we first hear the big news: Secretary of State Haig has resigned.

That must have caused a hubbub back home, I remark, and then we flick off the radio and concentrate once more on the scenery.