TRAVELING IN STYLE : Where Eagles Score : Capital tees: from hallowed St. Andrews to magnificent Mauna Kea, the best in fairways and greens.

Over the Valley of Sin they strode, red capes swirling in the North Sea breeze. Two students from the University of St. Andrews, Scotland, crossing the 18th fairway of a golf course. Not simply any course. Indeed not. It is the oldest in the world. To this unabashed romantic, sipping tea by a window in the Old Course Hotel, it was like flipping a page to the past. This is how, I mused, it must have appeared five centuries ago when goffer twosomes would sneak off to the links, daring to defy a ban on the sport imposed by King James IV. (So intently had his subjects been playing goff that they’d been neglecting archery practice--the national defense of the day.)

From this old gray town of St. Andrews, disciples have taken the game of golf to all climes. Even to the moon--if you count Alan Shepard’s wild swing on his space walk. It is from here, the forbidding citadel of the Royal and Ancient Golf Club (off limits to women), that the no-nonsense rules of golf are guarded. It is from this--dare I say?--rather plain-looking municipal course, designed and whipped by the forces of nature, that all the world’s golf courses take their inspiration--for bunkers named “The Coffins” and “Hell.” In ST. Andrews, tradition is everywhere--even in the open-to-the-sky ruins of the 14th Century cathedral where I stood the other day, bemused, before the tomb of Young Tom Morris. Morris was a beloved champion golfer of the 1870s, with the charisma of an Arnold Palmer. His grave stands out among all others due to a life-sized, quite joyful, bas-relief of Morris, dressed for a round with putter in hand.

Sooner or later, all true golfers feel the fervor of St. Andrews. Currently they’re saying that the Old Course--for the first time ever--may begin staying open on Sundays. Egad!

What is more likely to happen, local folks believe, is that the New (1894), the Eden and the Jubilee--three courses that flank the Old--will be made more attractive to visitors. Yes, change is coming to St. Andrews. It has already, with the sale of the Old course Hotel and its conversion to a country-club concept with pro shop, hot tub and swimming pool (indoors). But it still--and I doubt not, always--will bear the standard by which meaningful golf is measured.

So after St. Andrews, what?

I began thinking of top-class golf courses I’ve loved both for their tradition and their comfort. For one, the Pinehurst Hotel and CountryClub in the center of North Carolina. America’s golf mecca, they call it, because it owns and operates seven courses--more than any other hotel. Funny, because at Pinehurst, golf was an afterthought. Pinehurst was founded in 1895 as a health resort (complete with its own dairy herd for fresh milk), and the sports it offered were the ones then in fashion--polo, lawn bowling and archery. A few avant-garde guests, though, took to climbing the fences and disturbing the cows by trying the new sport of golf in the pasture.

Getting the hint, Pinehurst’s owner hired a Scotsman fresh from St. Andrews to lay out a golf course. The Scot did better than construct one course; he made four, and 400 more across the United States, during his lifetime in this country as world spread of his talent. His name was Donald Ross, a name that today virtually means tops in classic golf course design.

Courses at Pinehurst don’t have names, only numbers. And No. 2, nestled like the others within a forest of pines, ranks among the very best, with its angled fairways, ball-catching mounds and fall-away greens. It has been called demanding, delightful, difficult. What more could a golfer desire?

Courses No. 1, 4, 6 (by Tom and George Fazio) and 7 (by Rees Jones) are fine, too, but what really makes Pinehurst sing are the ways it caters to golfers. Breakfasts are cooked and brought super-fast, assuring players of being able to tee off 20 minutes ahead of their starting time. Friendly but vigilant rangers take over then, keeping play, in the Scottish-style, moving smartly.

After golf, however, all is leisurely Southern hospitality at the great white hotel with its wraparound veranda. I would hail a horse-and-carriage ride, or maybe walk, to the pretty New England-like village. Still, after dinner, I would succumb to more golf. (Films of famous old championship matches are shown every night in the lounge.)

MY THOUGHTS ALSO BEGAN to dwell on the Mauna Kea resort on the Kohala coast of Hawaii. Each time I visit, I’m as thrilled by the impact as I was the very first time I arrived. Stepping across the threshold (Hawaiian-fashion, there is no front door), I pause, staggered by the blue Pacific meeting the sky. My eyes drink in the flowers. My ears pick up the bird songs.

Now in its 23rd year, the Mauna Kea, originally a Rockresort but in the Westin hotel family since 1978, was the first to prove that a diamond of a golf course could be coaxed from impossible land. Out of ugly, black, crumbling volcanic lava, Robert Trent Jones created a magnificent landscape, so distinctive for its time that Arnold Palmer, Jack Nicklaus and Gary Player flew out to christen it with an exhibition.

In the beginning, guests who were accustomed to being pampered complained that the course was too tough. And so it was made easier; tees raised, bunkers erased, even the rough made smoother. But today’s golfers must be better sports, because the course has been restyled to reflect its original personality--relaxing pleasure from the front tees, a sinister challenge from the back. There being four tee choices to fiddle with, all guests are pacified.

When the game has ended, I retreat to the sugar-white, crescent-shaped beach, or simply stroll about the hotel, savoring its more than 1,000 art objects gathered from around the Pacific. Here a golden temple toy, there a priceless tapa. I love the sunset hour, too, when Hawaiian musicians set the evening’s mood by strumming old island melodies.

IF LEAN CUISINE IS IN at Mauna Kea, it’s the opposite at the Tryall Golf and Beach Club on Sandy Bay, Jamaica. I relish the memory of teatime there, served with all the proper silver and innumerable delicate (and fattening) tidbits. My senses reach out to the hotel, The Great House, perched high on a hill and furnished with handsome antiques, although the guest rooms are purposely maintained in an almost Spartan manner, so as not to clash with the brightness of the tropics. One goes to Tyrall for peace and serenity. It is the best tropical hideaway I know.

And its golf? Its course is a seaside jewel with lush fairways that weave through land where sugar cane and bananas once flourished. Along the seventh fairway an old, black cannon points to sea, guarding against pirates who never came. Just before the Christmas holidays each year, one of professional golf’s most appealing events occurs at Tryall--a team-play championship with leaders from the Senior PGA Tour coupled with the best women golfers.

SAILING IN ON A CLIPPER CRUISE ship is how I arrived at Sea Island Golf Club on St. Simon’s Island, Ga. It was the perfect way to be introduced, since it was by yacht that the club’s first golfers arrived in 1927.

The club is a former cotton plantation, with a clubhouse that once was a horse barn. It is built of that mellowed stuff they call tabby--a mixture of oyster shell, lime and sand. I could have spent all day studying the photographs that stud its walls. All of golf’s heroes who played Sea Island are there, from the legendary Bobby Jones, one of its earliest admirers, to Davis Love III, a sensational young touring pro of today. I adore each one of the 36 holes, and their old-fashioned names--”Teaser” and “Splasher” and “Dare You.”

Both Sea Island and its neighboring St. Simon’s Island Golf Club are owned by the Cloister Hotel, and what a delicious experience my visit turned out to be. Ballroom dancing is featured after dinner, and a corps of local residents volunteer men to dance with the single ladies. Forty-four fireplaces dot this Spanish-looking inn with guest rooms that are enormous in the best “Gone With the Wind” manner--gaily decorated, with ceiling fans for effect.

Staying at The Cloister is a reassuring reminder that the art of gracious service survives. Yet, the hotel has been willing to bend with the times. By popular request, gentlemen no longer need to wear ties to breakfast. Good-looking golf sweaters will do.

THEN THERE IS BANFF SPRINGS Hotel in Banff National Park in the het of the Canadian Rockies--the most spectacular of the golf courses I know.

Spread along the Bow River, shaded by pine trees, the course seems to always have been there, although it is a bit disconcerting to line up one’s shots with mountain peaks.

The course was built in 1911 by the Canadian Pacific Railway with nine holes, and it was expanded to 18 by German prisoners of war during World War I. The hotel--with turrets that are often seen on picture postcards--recently was refurbished from top to bottom and now appears even more glamorous than ever.

AT TRYALL, AN OLD BLACK CANNON POINTS TO SEA, GUARDING AGAINST PIRATES WHO NEVER CAME

Meals at Banff are generous and memorable. Rooms feature high ceilings, and the occasional sighting of bears sets a mood that says this is a true wilderness and a pleasure for the golfer.

What price Top-Class Golf?

St. Andrews: Old course, $25; other three, $8-$12. Caddie $15, bag carriers $12. Old Course Hotel & Country Club rates: $111 single, $177 double, breakfast included, season (May-November); off season, $68 single, $114 double.

Pinehurst: $45 No. 2 course, $35 No. 7, others $25; cart $22. Two-night, three-day golf package, $234-$339 per person, breakfast and dinner included, but a $10-$20 surcharge to play No. 7 or No. 2. Rates without golf, $163 single, $96 per person double, breakfast and dinner included.

Mauna Kea: $34; cart, $24. Unlimited golf package (four-night minimum) May to mid-Dec., $220-$310 per day single or double, without meals. For breakfast and dinner on the golf package add $50 per person per day. Regular rates without golf but with meals, $198-$278 per person.

Tryall: Golf and all meals are included in high-season rates (through Easter) of $230-$280 single, $260-$310 for two. A villa with servants costs $1,900 a week, with golf extra at $25 plus caddy or cart. Lower rates after May.

The Cloister: $26, plus $20 for caddy or cart to play Sea Island or St. Simon’s Island Golf Club. Unlimited golf package, mid-March through May, $144-$196 per person per day, all meals included. Package is slightly less June-November, and much less in winter. Spring rates without golf but with meals are $60-$112 single, $100-$152 for two.

Banff: Gold (May-October only) about $10, plus cart or caddy. Rates, $104-$125 per room.

Rates are subject to change. All courses are among the world’s best 100 as rated by Golf Digest.