She spent her early childhood in Phillipsburg, New Jersey, where her father was an attorney practicing with future New Jersey governor Robert B. Meyner. In 1936, her father died of a heart attack. In 1939 her mother married sales engineer Harry Lawrence Peers, and the family moved to Dallas, Texas, where she was known as Vera Jayne Peers. As a child she wanted to be a Hollywood star like Shirley Temple. At age 12, she took ballroom dance lessons. She graduated from Highland Park High School in 1950. While in high school, Palmer took violin, piano, and viola lessons. She also studied Spanish and German. She consistently received grades in the high Bs including in mathematics.