IN St. Louis, the dog days of summer give way to a fall as glorious as anywhere else on Earth: Days are warm, sometimes even hot, and nights are cool. September ushers in a long Indian summer that often lasts till Thanksgiving. It offers an ideal time to spend a week or a long family weekend in the Gateway City. Here are some of the sights and activities to consider: The Arch. The Gateway Arch, 630 feet of stainless steel towering above the grounds of the Jefferson National Expansion Memorial, is an ideal place to begin. Like the Vietnam Veterans' Memorial in Washington, D.C., the arch's surface invites the laying on of hands. Even in midsummer, it is cool and comforting, and so is the underground museum beneath it, the exquisite Museum of Westward Expansion. Life-size displays include a teepee, a sod house, and a beaver pond. Photos, diary excerpts, and displays of clothes, tools, and weapons tell the story in the pioneers' own terms. Informative talks are given throughout the day at various focal points. The recent imposition of entrance fees to national parks includes the arch; the maximum is $3 per family. The trip up the arch by electric car costs an additional $2.50 for adults, 50 cents for children.
River trips. The park grounds outside the arch make a good place to munch lunch while watching barge traffic move along the mighty Mississippi. It's not unheard of to dip a toe into the river here, but wading is strongly discouraged.
The best way to see the river itself is by boat. The President, the city's largest river cruiser, will be in Cincinnati in October for that city's bicentennial, but three small paddle-wheeler replicas will continue cruising, and the President will be at home here during the rest of this month and again in November. All the boats make three-hour sightseeing cruises (priced at $8.50 for adults and $3.75 for children) and five-hour day or evening cruises ($27.50 and $13.50), including a buffet. The now-stationary Admiral - once the art nouveau queen of excursion boats - offers family entertainment and food by day, and music and dancing by night (no admission fee, but charges for attractions inside).
Exploring downtown. A day in the heart of the city could include a walk on the cobblestone street of Laclede's Landing, the renovated warehouse district on the original settlement site, where Pierre Laclede's birthday is celebrated throughout this month.
West of the arch and past the impressive public library building is Campbell House (admissions $2 and 50 cents), a mid-Victorian town house with the original furnishings. But children will find more excitement nearby at the renovated Union Station. Boutiques now replace trains, but there's also fun and plenty of interesting railway history. South of the arch, past the Cardinals' baseball stadium, is the boyhood home of poet and journalist Eugene Field (1850-95), which is now a toy museum ($17 and $8.50) and the departure point for open-air tram tours ($20 and $5), with stops downtown and in Forest Park.
Forest Park. This site of the 1904 World's Fair can keep visitors busy for at least a day. The institutions it contains - a zoo, art museum, and history museum - are inexpensive or free. The St. Louis Zoo abolished outdoor cages in the 1930s in favor of habitat duplication with natural barriers. Since 1970 it has continuously upgraded its display environments for birds, reptiles, small and large primates, and major cats; it has also opened a Children's Zoo (45 cents) and added a spiffy child-size railroad.
Forest Park also contains the city's art museum, a local-history museum, a floral display house, and an expanding St. Louis Science Center, definitely child-oriented.
An outdoor playground demonstrates scientific principles, and two model dinosaurs capture most kids' attention instantly. Inside, visitors find a science laboratory, star shows, and a discovery room for hands-on learning. Fees range from 50 cents to $3.
McDonnell Douglas's Aerospace Center Museum near Lambert International Airport also welcomes visitors.
Away from the center. Moving outside the downtown area, families will enjoy the Missouri Botanical Garden (Shaw's Garden), Grant's Farm, and the Jefferson Barracks Historic Park.
The garden, a gift to St. Louis by one of its early philanthropists, is the largest in the Midwest ($2 for adults, no charge for children).
Tours at Grant's Farm, a 281-acre homestead once own by Ulysses S. Grant, include refreshments, a bird show, a ride through the antelope and bison park, and a stop at Grant's ``Hardscrabble'' cabin. They are available through mid-October (no charge, but reservations are required).
Regular Wednesday-through-Saturday tours show off the restored buildings at Jefferson Barracks, an Army post that was active from 1826 to 1946.
Farther afield are unusual sites that educate as well as entertain: Laumeier Sculpture Park, Ralston Purina's Research Farm, the Shaw Arboretum (a good picnic place before or after the research farm), Cahokia Mounds (an archaeological trove of Indian prehistory), and the National Museum of Transportation.
The transportation museum is a wonderful place for children to let off steam as they scramble over and through railway history. Purina Farm, with its delightful multi-image theater, animal barn, and pet center, is en route to Six Flags Over Mid-America in Eureka, Mo.
When it's time for education to give way to recreation, try a day of pure entertainment at Six Flags, open weekends through October ($12 to $17, with discounts available). Autumn generally means smaller crowds and shorter lines.
That's St. Louis's family entertainment off-season. If you miss the fall, most attractions start up again in April to take advantage of the long, sparkling mid-country spring. Practical information
For more details, phone the St. Louis Convention and Visitors Bureau at (314) 421-1023.