"serve apprenticeship" is correct and usable in written English
You can use it to refer to a period of time when someone takes on a job or task in order to learn specific skills or knowledge related to that job or task. For example: "He served his apprenticeship at a local mechanical workshop for two years before becoming a qualified mechanic."
A school-business partnership allows upper-grade students to serve apprenticeships.
He later served apprenticeships in stock companies.
After serving apprenticeships as a millwright and a ship modeler, he went to London, where he worked and studied under the Scottish engineer John Rennie.
Few if any of these young men have served apprenticeships with the Telephone Company, which the police eavesdroppers regard as the only worthwhile alma mater.
He then turned to boat building as a career, serving apprenticeships with Lindsay Lord, a premier designer of recreational power boats, and John Hacker, a leading designer of racing boats.
Having been inspired by Sonny Rollins, John Coltrane and Rahsaan Roland Kirk in the 60's and having served apprenticeships with Cecil Taylor and Andrew Cyrille in the 70's, he comes by a big, discursive style naturally.
If electric quilts and flocked wallpaper are common themes in current Philadelphia artwork, they might have something to do with the Fabric Workshop and Museum, where Marti and Da Corte have served apprenticeships.
Kirk Douglas, Anthony Quinn, Robert Mitchum, Jack Palance and Lee Marvin were among the postwar stars who served apprenticeships — some long, some short — as outlaws gunned down in the last reel of westerns or as hoodlums crumpling under police fire in crime pictures.
In the United States, prominent chefs like Jean Joho, of Everest in Chicago, Hubert Keller, of Fleur de Lys in San Francisco, and Mr. Vongerichten, proprietor of a dozen New York restaurants, served apprenticeships at Auberge de l'Ill.
The Rangers wanted Mottau to serve an apprenticeship at Hartford.
The idea of having to serve that apprenticeship made me nervous.
