AMERICA'S VAGABOND youth who invaded Europe in the '70s on $5 and $10 a day found out that travel doesn't have to be expensive to be fun. That's true even today for tourists determined to explore the nation without spending a bundle.

If luxury isn't in your budget, forget the $50- or $100-a-night hotel and motel charges. For a whole lot less you can enjoy a vacation at some of the country's best beaches, visit the World's Fair in Knoxville, splash in mountain lakes or sightsee in the big cities.

In these economically troubled times, the travel industry increasingly is aiming at the budget-minded traveler. When you hit the road this summer, consider these alternatives to high-cost lodging:

* Stay in an American Youth Hostel: Despite the name, America's 300 coast-to-coast hostels are open to members of any age at $4 to $8 per person per night.

On Nantucket Island, an 1873 life-saving station has been converted to a hostel. In San Francisco, the hostel at Fort Mason Park gives you a clear view of the Golden Gate Bridge. Admittedly, the facilities are spartan--you get a bunk bed in a bunk room, though some family rooms are available, and you are expected to perform a clean-up chore.

A handbook of accommodations is free to members. Single membership, $14; family, $21. For information: American Youth Hostels, 1332 I St. NW, Washington, D.C. 20005, 783-6161.

* Go camping in a national or state park: The National Park Service maintains campsites in 103 parks and other areas, which means you could probably take a cross-country trip staying only in America's most scenic spots for $4 to $6 a night per car, plus park entrance fee.

In most parks, registration is first-come, first-served, but reservations through Ticketron ($2 fee) can be made for: Cape Hatteras National Seashore, N.C. (736 campsites); Shenandoah National Park, Va. (702 campsites); Grand Canyon National Park, Ariz.; Great Smoky Mountains National Park, N.C.-Tenn.; Rocky Mountain National Park, Colo.; Sequoia-Kings Canyon National Parks, Calif.; and Yosemite National Park, Calif.

For a directory, "Camping in the National Park System" (No. 024-005-00807-1), send $2.25 to the Government Printing Office, 710 N. Capitol St., Washington, D.C. 20402. For state park information, contact the tourism office in the state capital.

One camping couple from Washington always aims for a location where two or three camping areas are nearby in case their first choice is filled.

* Try Kampgrounds of America, or other private campsites. KOA, the largest network, has 730 locations, including 56 in Canada. The average cost is about $9 to $10 a night, depending on family size and whether water, sewage and electrical hookups are desired. Three-fourths of the campgrounds have swimming pools. Popular for recreational vehicles, though tenters make up about 20 percent of the business.

For the world's fair, KOA has added rough-lumber and log cabins to nine Knoxville-area campgrounds, sleeping six for $36 on bunkbeds with mattresses. Bring your own sleeping bag.

Reservations must be made through individual campgrounds. A free 100-page directory is available from the campgrounds, or send $1 to KOA, P.O. Box 30558, Billings, Mont. 59114.

* Join the "bed and breakfast" trend: An idea the vagabonds brought home from Europe, you stay in private homes at substantial savings. An estimated 40 to 50 groups have formed since 1978, representing 3,000 to 4,000 host families nationwide.

Two of the largest: Bed and Breakfast League, headquartered in Washington, with about 165 host families in 100 cities. For $25 family membership, you get access to the host list and a central reservation number. Room rates are about $30 to $38, with continental breakfast. For information: 232-8718.

Also, Bed and Breakfast International of Kensington, Calif., with 150 hosts concentrated in the San Francisco Bay Area and elsewhere in California. No membership fee, with double rooms averaging $34 to $44, including a full breakfast. For a brochure, call: (415) 525-4569 or (415) 527-2928.

* Seek out a budget- or economy-class hotel or motel: These stripped-down models appear to be undergoing a building boom. There are several large chains in the United States, which claim to beat intermediate-priced lodgings such as Holiday Inns by 20 to 30 percent.

The cheapest, by industry agreement, appears to be Motel 6 of Santa Barbara, Calif., with 329 motels in 38 states. Small rooms, no phones, no on-premise restaurant, pay-TV, but the standard rates are $18.45 double and $21.45 for a family of four. Twenty-five to 35 new motels opening a year. For a directory, write Motel 6, 51 Hitchcock Way, Santa Barbara, Calif. 93105.

With larger rooms and more amenities--and somewhat more expensive--are: Days Inns of Atlanta, 320 hotels, averaging $32 a night for a couple. Free color TV. Toll-free reservations: (800) 241-7200; Econo-Travel Motor Hotels/Econo Lodge of Norfolk, 150 motels mostly in the East, averaging $25 for a double in 1981 (800) 446-6900; Rodeway Inns of Dallas, 145 motels, mostly on the West Coast and the Sunbelt but expanding to the Northeast (800) 228-2000; Super 8 of Aberdeen, S.D., 121 properties but adding a new one every nine days, mostly in the central part of the country. Lowest single, $14.88 in Miller, S.D., and highest, $49.88 in West Palm Beach, Fla. (800) 843-1991.

* Swap your home: The Vacation Exchange Club, formed in 1960, claims 6,000 listings in the United States and around the world. To be included in the directory, the fee is $21. To purchase the directory only, $14. After that, it's up to you to make the swap for a home or apartment in a place you want to visit. For an application, write: Vacation Exchange Club, 350 Broadway, New York City, N.Y. 10013.

* Rent a recreational vehicle: Not exactly cheap, especially if you are driving long distances, but you can save by preparing your own meals. Americar RV Rental System of Alexandria charges $325 a week, plus 12 cents a mile, for a four-person Shasta that includes stove, refrigerator, shower and toilet. Gas is extra (at about 12 miles to the gallon) and so are campground fees.

One firm that handles out-of-town rentals is American Safari of Miami, with 61 outlets nationwide. Rates range from $365 a week (15 cents a mile) for a three-person RV to $860 a week (1,500 free miles) for a large deluxe model sleeping eight. Reservations: (800) 327-9668.

* Haul a camping trailer: A very basic four-person trailer-tent rents for $50 a week from 7 Corners Rentals in Falls Church. King Rentals of Rockville charges $125 a week or $375 monthly for trailer tents equipped with stove, sink and cooler in models sleeping four, six or eight.

* Stay with friends and relatives: The old standby, it's budget travel at its cheapest. Remember the thank-you gift, or take your hosts out to dinner to keep this option open.

Washington train buff John Hickman, a Veterans Administration public relations specialist, has scheduled four weekend trips to New York City this summer in his privately owned, stainless-steel observation car and lounge, "No. Forty-Eight."

Built in 1948, the car originally ran on New York Central's "Pacemaker" and "James Whitcomb Riley" in the Midwest. It has since been reconditioned to the style of first-class travel in the '40s and '50s. It seats about 40.

At $210 a couple or $130 single, the fare includes roundtrip transportation (the car is attached to a regularly scheduled Amtrak train), complimentary beverages and snacks, continental breakfast going, a buffet dinner returning, and Saturday night accommodations at the Wellington Hotel.

Departures at 8:30 a.m. from Union Station on June 26 and July 3, 10 and 17, leaving New York at 5 p.m. on Sunday. For information: Forty-Eight to New York, 13014 Atlantic Ave., Rockville, Md. 20851. Enclose stamped, self-addressed envelope.