Goldman was born in Oakland, California in 1944 and grew up in Watsonville. As a youth, he studied piano and enjoyed model-making and drawing. Before devoting himself entirely to the arts, he served as an electronics technician in the United States Air Force. He received his Associated Arts Degree from Cabrillo College near Santa Cruz, California, and in December 1971, he graduated with a Bachelor of Fine Arts Degree in Life Drawing from the University of Hawaiʻi.

In early 1972, he began his animation career when he joined Walt Disney Productions. His first assignment was as an In-betweener to Frank Thomas on the film Robin Hood. He then worked alongside Don Bluth as an Animator on Winnie the Pooh and Tigger Too and The Rescuers. He was a Directing Animator on Pete's Dragon, The Small One, and The Fox and the Hound.

Goldman and Bluth began to probe the possibility of creating their own animated projects. United by the same goal of restoring the lost techniques of classical animation, Goldman, Bluth, and animator John Pomeroy produced (in Bluth's garage) the classically animated television special, Banjo the Woodpile Cat. This won the National Film Advisory Board Award of Excellence and the Golden Scroll Award from the Academy of Science Fiction, Fantasy, and Horror Films. It was completed in 1979. That same year, Goldman, along with Bluth and Pomeroy, resigned from Walt Disney Productions to establish themselves independently.

Under their independent banner, they have produced several feature films, starting with The Secret of NIMH (1982), which won the Saturn Award for Best Animated Feature from the Academy of Science Fiction, Fantasy, and Horror Films, and An American Tail (1986), which at that time, earned the biggest box office gross for the first release of an animated film in motion picture history, as well as two Grammy awards for the song Somewhere Out There. They also created the animated laserdisc video games Dragon's Lair (1983) and Space Ace (1984). Today, these games remain popular among video game enthusiasts.

In 1986, Goldman and Bluth moved their entire operation, including 87 employees and their families, to Dublin, Ireland at the invitation of the Irish Government. Their third feature film, The Land Before Time, was their first production finished entirely in Ireland. It was released by Universal Pictures during Thanksgiving 1988. It celebrated a record-breaking opening weekend gross for an animated film of $7.5 million. This was followed by All Dogs Go to Heaven (1989). Some of their most recent projects under the Don Bluth name include Rock-A-Doodle (1990), Thumbelina (1993), and A Troll in Central Park (1993).

Goldman is a member of the Academy of Motion Picture Arts and Sciences, is on the Board of Trustees of the Phoenix Art Museum, and is a member of the United Way, Alexis de Tocqueville Society. He and creative partner Don Bluth most recently produced the feature film Anastasia, which was released in November 1997, and Titan AE, which was released in June 2000.