Yetmgeta artistic practice started at an early age. At fifteen years of age, he won first prize in a national art competition, known as the "All Ethiopian School" competition. After completing high school, he attended the Alle School of Fine Arts and Design in Addis Ababa (1963-1968). While attending the university, Yetmgeta shared a studio with Alexander Boghossian, known as Skunder Boghossian. Boghossian, along with Gebre Kristos Desta and Karl Heinz Hansen (his professors at the School of Fine Arts), have been cited as major influences in the artist’s development. Yetmgeta has taught at the Alle School of Fine Arts and Design since 1978.
In the aftermath of the 1974 Ethiopian Revolution, when the Emperor Haile Selassie was overthrown, many artists went into exile. The socialist military rule of the Provisional Military Administrative Council (known as the Derg) that followed, established repressive practices toward the arts, mandating the use of social realism and censoring subject matter. Yetmgeta, however, chose to stay in Addis Ababa. Since the end of Derg rule, Yetmgeta has enjoyed international prominence and has exhibited widely.
On 24 August 2019, the Ethiopian Cultural Garden, featuring Yetmgeta's original design in addition to a replication of his work "When the Sun Meets the Moon", was dedicated in Rockefeller Park in Cleveland, Ohio. It is the first cultural garden from the continent of Africa and it is believed to be the only monument to Ethiopia in the United States.