Entrance fees at most European museums aren't budget-breaking, but when you and the rest of the family are trying to see as many as you have time for, the total cost can add up quickly. So can subway and bus fares as you crisscross a city to take in its major sights.

These are the so-called incidentals of travel, which -- depending on how active a sightseer you are -- can become a substantial expense.

A good way to cut these costs is to take advantage of the variety of special discounts available to transatlantic travelers. They often come in the form of a card purchased for a fee that provides for free or reduced admission to the museums and other attractions of a city or country or unlimited travel for a specified time period.

In Norway, for example, the Oslo Card is good for free bus and trolley fare within the capital; free admission to museums and historic sites; discounts on selected tours and reduced rates for the rental of a car, bicycle and even a sailboard. The cost of the card, sold at Oslo hotels and tourist information offices, is about $6.50 per adult for one day and $13 for three days. Children 11 and under get the card at half price.

Elsewhere in Europe, there are also discounts designed for seniors, for students and for transportation by air, rail, bus and ferry.

Some discount cards can be obtained only by purchasing them before you leave the United States; others are readily available in the city or country you are visiting. Obviously, they are most valuable if you plan to do a lot of sightseeing and museum-viewing and if you generally get around a city or country on public transportation.

Among the discounts available in major European destinations:

*The Netherlands: There are two separate discount cards for travel in the Netherlands. The newest, offered for the first time last month, is the Holland Leisure Card, aimed at the general sightseer. The Holland Culture Card, offered for the past decade, is more suited for travelers interested in visiting the country's many museums and attending concerts and other performing arts events.

Both cards include a 55 percent discount on a first-class one-day pass for unlimited travel by rail throughout the country. This is a particularly good bargain for visitors who want to make Amsterdam their headquarters and see other parts of Holland on one-day round-trip excursions. Similar discounts are offered on bus, tram and subway fares throughout the country, as well as 25 percent discounts on boat and motorcoach sightseeing tours.

The Leisure Card, which sells for $7.50 per person and is good for a year, provides half-price discounts to 18 top tourist attractions; free admission to Holland's four major casinos; a 25 percent discount on car rentals and a 10 percent discount on purchases at a major department store and its branches. The card comes with a Holland tourist map and a calendar of major events.

The Holland Culture Card, selling for $15 and also valid for a year, includes a National Museum Card -- good for free (and unlimited) admission to 250 museums. The culture card also permits the holder to reserve regular-priced tickets for concerts, theater, opera and ballet from blocks of seats held for travelers from abroad. The tickets are sold at tourist information offices in Holland, and no additional fee is charged for the reservation service.

Both the culture and leisure cards can be purchased from a U.S. travel agent or the Netherlands Board of Tourism in New York. They are not available in the Netherlands.

For more information: Netherlands Board of Tourism, 576 Fifth Ave., New York, N.Y. 10036, (212) 245-5325.

*Great Britain: The current "Treasure Houses of Britain" exhibit at the National Gallery has focused attention on Britain's "Open to View" ticket, good for admission to 500 homes, castles, museums, gardens and other historical buildings, including the Tower of London and the crown jewels. The ticket costs $23 for adults and $11.50 for children ages 15 and under and is valid for one month from the day it is first used. It can be purchased through a travel agent before you depart for Great Britain.

BritRail, the British rail system, is offering visitors who are 65 or older a first-class pass for a week's travel on the British railway network for the cost of a second-class pass. A first-class pass for seven days of travel is $155, but seniors get it for $115. These and other rail discounts are sold in the United States by travel agents and BritRail Travel International, 630 Third Ave., New York, N.Y. 10017, (212) 599-5407.

Another inexpensive way to see England, Scotland and Wales is by bus. Britain's Express Coach Network offers a $10 Britexpress Card to overseas visitors, which is good for one-third off standard fares for a 30-day period. The card can be purchased from a travel agent or at major bus ticket offices in Great Britain.

If you are staying in London, consider the London Explorer Pass, which provides unlimited travel on the city's big red buses and excellent subway system. A three-day pass is $12 for adults and $5 for children (ages 15 to 5); four days, $15 for adults and $7 for children; seven days, $20 for adults and $10 for children. These passes are available at subway fare booths in London.

Other discounts in Great Britain are detailed in a pamphlet, "Tourist Tickets and Discount Cards." For a copy: British Tourist Authority, 40 W. 57th St., New York, N.Y. 10019, (212) 581-4700.

*France: The French National Railroad has introduced a new discount pass for rail travel in the country that provides more flexibility that do most European rail passes. The new pass, costing $130, is good for any nine days of second-class rail travel within a one-month period.

The advantage is that pass-holders can stop over at towns and cities on their itinerary for several days without using up valuable travel time. The standard rail pass usually is good for a one-, two- or three-week period, and every day counts against you, whether or not you are traveling. A standard seven-day, second-class French rail pass is $115.

The rail passes also are good for two free days of travel on the Paris Metro, the easy-to-use subway system; free transfer from the Paris airport into the city; and discounts or free admission to many museums. They can be purchased before you depart from a travel agent or the French National Railroad, 610 Fifth Ave., New York, N.Y. 10020, (212) 582-2110.

Also available for seniors is the Carte Vermeil, which provides for a 50 percent discount on rail travel for males 62 and older and females 60 and older. This card can be obtained by filling out a special form at a railway station. The cost is about $8 for a one-year period. The card is not good during weekend rush periods or over holidays.

For more information: The French National Railroad (see above) or the French Government Tourist Office, 610 Fifth Ave., New York, N.Y. 10020, (212) 757-1125.

*Ireland: The "Passport to Ireland's Heritage and Culture" is a booklet containing discount admission tickets to a number of selected historic homes, gardens and castles in Ireland, including many of the country's major tourist attractions. The booklet is $5 and the discount is about one-third off each entrance fee. They can be purchased at tourist information offices within Ireland.

For more information: Irish Tourist Board, 757 Third Ave., New York, N.Y. 10017, (212) 418-0800.

*The Scandinavian capitals: Along with the Oslo Card, travelers to Scandinavia also can purchase a Copenhagen Card and a Stockholm Card. Both cards, available at hotels and tourist information offices in each city, are good for free unlimited public transportation during the card's validity period and admission to many museums.

The Copenhagen Card, for example, gets you into Tivoli Garden, the Danish capital's fairy-tale amusement park, at no extra charge. In the Swedish capital, the Stockholm Card is good for entrance to the Wasavarvet, a very popular museum where what is believed to be the world's oldest identified ship, the Wasa, is displayed. Built in 1628, the ship sunk outside Stockholm on its maiden voyage and remained lost until 1956.

A one-day adult Copenhagen Card is about $6; two days, $11; and three days, $15. A Stockholm card is slightly more. A child's card (ages 5 to 11) is half-price.

For more information: The Danish, Norwegian or Swedish Tourist Board, 655 Third Ave., Suite 1810, New York, N.Y. 10017, (212) 949-2333.

The discounts described above are only samples of what is available. Before leaving the United States for these or other European countries, travelers should consult a travel agent or the country's national tourist office in New York for any other discounts available in their particular age group. In many cases, it may be too late to apply once you have arrived at your destination.

*IN JEFFERSON'S FOOTSTEPS: To mark the 200th anniversary of Virginia's State Capitol, the Historic Richmond Foundation is sponsoring a two-week trip to Paris and southern France "to study buildings that inspired the capitol's designer, Thomas Jefferson."

The capitol, as the foundation points out, was "the first major building in the form of a classical temple to be built in Europe or America since antiquity." It was constructed while Jefferson was a U.S. minister in France, from where he sent plans and a model to Richmond.

The foundation's tour, departing March 19 from Washington/Dulles, follows in Jefferson's path from Paris to Provence at the same time of year Jefferson began a three-month exploration of France and northern Italy. On the itinerary is, of course, Nimes, where a special tour has been arranged of the Maison Carre'e, the Roman temple that was "the grand inspiration" for the Richmond capitol.

The tour is $3,620 per person (double occupancy), which includes round-trip air fare to Paris; 13 nights lodging (including five nights in Paris at the Hotel de Crillon; four nights in Avignon; and four nights in Carcassonne); continental breakfasts; some lunches and dinners; and a $100 tax-deductible donation to the Historic Richmond Foundation.

For more information: Dynasty/Red Carpet Travel Service, 701 E. Franklin St., Richmond, Va. 23219, (804) 644-4631.