'set out on a quest' is correct and usable in written English. You can use it to describe beginning a journey or task with a goal at the end. For example, "The brave adventurers set out on a quest to find the enchanted artifact deep in the forest.".
From there, we set out on a quest to get 100 photographs of Amanda, with the hope of one day putting together an exhibition.
I've set out on a quest to try to improve how we fight as a family.
Dr. Eisenberg, a general practitioner with no particular expertise in research or dermatology, set out on a quest to save his son.
Adam and Neil Pearson, identical twins who share the same genetic disease, neurofibromatosis 1 (Nf1), set out on a quest to find out why they have different symptoms.
The characteristic Kiarostami hero is a man who sets out on a quest and arrives nowhere.
Janine junks her job as a maths lecturer and sets out on a quest for answers.
When his beloved friend dies, Gilgamesh sets out on a quest for immortality.
Since I tried Ludwig back in 2017, I have been constantly using it in both editing and translation. Ever since, I suggest it to my translators at ProSciEditing.
Justyna Jupowicz-Kozak
CEO of Professional Science Editing for Scientists @ prosciediting.com