I'll start my list with some classics that you, the parent, may already know - you'll see plenty of Rodgers and Hammerstein on this list, as well as Lerner and Lowe and Kander and Ebb, and some names you may not recognize unless you're a theatre buff. (Please note that I have avoided Disney shows, mainly because your kids probably already know 90% of the music from their shows.) But I'll follow it up with some more contemporary composers, plus some terrific songs from shows that were...well, not so terrific.
Classics from the Golden Age of Theatre
"The Stepsisters' Lament" from Cinderella (Rodgers and Hammerstein)
"The Brotherhood of Man" from How to Succeed in Business Without Really Trying (Frank Loesser)
"I Cain't Say No" from Oklahoma (Rodgers and Hammerstein)
"Shy" from Once Upon a Mattress (Mary Rodgers)
"Do-Re-Mi" from The Sound of Music (Rodgers and Hammerstein)
"Tevye's Dream" from Fiddler on the Roof (Bock and Harnick)
"Fugue for Tinhorns" from Guys and Dolls (Frank Loesser)
Other great numbers to check out:
"Anything You Can Do" from Annie Get Your Gun (Irving Berlin) - also "Doin' What Comes Natur'lly," "You Can't Get a Man with a Gun," "There's No Business Like Show Business," "Who Do You Love, I Hope", and "I Got the Sun in the Mornin'".
"I Won't Grow Up" from Peter Pan (Charlap, Styne, Leigh, Comden and Green) - also "I Gotta Crow," "I'm Flying," and "Hook's Tango".
"I Don't Need Anything But You" from Annie (Strouse and Charnin) - also "It's the Hard Knock Life," "I Think I'm Gonna Like It Here," "Easy Street," "You're Never Fully Dressed Without a Smile," and "NYC".
"Getting to Know You" from The King and I (Rodgers and Hammerstein) - also "I Whistle a Happy Tune," "March of the Royal Siamese Children," "A Puzzlement," "Shall I Tell You What I Think of You?", and "The Small House of Uncle Thomas".
"There Once Was a Man" from The Pajama Game (Adler and Ross) - also "Racing with the Clock," "Once a Year Day," "Steam Heat," "Think of the Time I Save," and "7-1/2 Cents".
"I am the Very Model of a Modern Major-General" from The Pirates of Penzance (Gilbert and Sullivan) - also "Climbing Over Rocky Mountain," "When the Foeman Bares His Steel," and "With Catlike Tread".
Contemporary/Lesser Known Songs
"What a Remarkable Age" from Titanic (Maury Yeston)
"Twenty Million People" from My Favorite Year (Ahrens and Flaherty) -
"On the Steps of the Palace" from Into the Woods (Stephen Sondheim) - Prologue
"Here I Am" from Dirty Rotten Scoundrels (David Yazbek)
"The Sad Tale of the Beauxhommes" from Once On This Island (Ahrens and Flaherty)
"A Trip to the Library" from She Loves Me (Bock and Harnick)
"When I Get My Name in Lights" from The Boy from Oz (Peter Allen)
"Little People" from Les Miserables (Schoenberg and Boublil)
"Seize the Day" from Newsies (Menken and Feldman)
"Neat to Be a Newsboy" from Working (multiple)
Other contemporary songs to check out:
"What It Means to be a Friend" from 13: The Musical (Jason Robert Brown)
"We Want to Take the Ladies Out" from Just So (Drewe and Stiles) - also "There's No Harm in Asking," "The Parsee Cake Walk," and "Please Don't Touch My Stove".
"The Rum Tum Tugger" from Cats (Andrew Lloyd Weber) - also "Mungojerrie and Rumpleteazer," "Growltiger's Last Stand," "Macavity: The Mystery Cat," "Skimbleshanks, the Railway Cat," and "Mister Mistoffelees".
"Oh My God, You Guys" from Legally Blonde (O'Keefe and Benjamin) - also "So Much Better," "Whipped Into Shape," and "Bend and Snap".
"My Strongest Suit" from Aida (John and Rice) - also "Fortune Favors the Brave," "The Gods Love Nubia," and "Every Story is a Love Story".