If homespun fun is your pleasure, then Arkansas and Tennessee are the places to be in 1986. Vancouver hosts a world's fair this year; the Statue of Liberty gets a birthday party; Denmark invites fairy-tale fans, and stargazers, amateur and professional, are heading south of the Equator for the best views of Halley's comet. The new year's big travel events include:
*Liberty Weekend, New York City: It's a gala celebration marking the centennial of the dedication of the Statue of Liberty as well as completion of the restoration work begun in 1984. As the opener, President Reagan will relight the torch on July 3. A full weekend program follows, including major fireworks displays, a concert in Central Park, a "tall ships" parade up the Hudson River and 5,000 new Americans being sworn in as citizens en masse on Ellis Island. July 3 through 6.
*Tennessee Homecoming '86: A statewide festival of "Tennesseans celebrating being Tennesseans -- part reunion, part history lesson and part good old-fashioned hoe-down." February through December.
*Arkansas Sesquicentennial: A statewide festival with at least 500 scheduled events will celebrate Arkansas' 150th birthday. Among the highlights: the Sesquicentennial Wagon Train, tracing a 1,500-mile route from Fayetteville in March to Little Rock in June, and the Salute to Statehood Weekend, June 13 to 15 in Little Rock. Yearlong.
*Expo 86, Vancouver, British Columbia: The big travel event of the year is the world's fair in Vancouver, a spectacularly scenic city. The theme is "World in Motion -- World in Touch," and officials hope to attract 8 million visitors. May 2 to Oct. 13.
*Halley's comet: The comet can be seen practically worldwide, but the best seats are in the Southern Hemispere. Travelers have a choice of comet-viewing cruises or tours to the Caribbean, South America, Africa, the South Pacific and Australia. Through the end of May.
*Hans Christian Andersen Year, Denmark: The Danish author published his first fairy tale 150 years ago, and the whole country is honoring this anniversary with readings of his tales, a festival of his plays, exhibits and "fairy-tale tours." Yearlong.
*Treasures of Mount Athos, Thessaloni'ki, Greece: A community of 20 Eastern Orthodox monasteries in northern Greece, the region of Mount Athos is off-limits to women and open to men only by special permit. So its rare and ancient treasures -- manuscripts, gold and silver relics, icons -- are seldom seen by the public. This year they will be displayed in conjunction with Philoxenia-'86, an international tourism trade fair in nearby Thessaloni'ki. April 23 to 27.
*125th anniversary of the start of the Civil War, Virginia: A variety of special events at Virginia's many Civil War sites will commemorate the anniversary, including a national ceremony at Manassas National Battlefield Park, July 19 to 21. Elsewhere in Virginia, from mid-June through Labor Day.
*Sun Valley's golden anniversary, Idaho: The famous ski and summer resort is throwing a big party for its 50th birthday, with special celebrity ski races, a film festival, snow sculptures and other winter as well as summer events. Yearlong.
*Heidelberg University's 600th anniversary, West Germany: West Germany's most famous university, founded in 1386 and locale for the popular operetta "The Student Prince," celebrates its 600th birthday with a year of special exhibits and cultural programs. The highlight is Festival Week, Oct. 12 to 19, featuring concerts, pageantry and other events. Yearlong.