I think we can call this sub-genre quest and it covers all the details in your question. Filmsite.org mentions quest as a sub-genre of Action/Adventure genre also. Additionally, it can be a plot device, a trope or a symbol in literature.
Quests, an immemorial trope in literature, are common in fantasy. They can be anything from a quest to locate the MacGuffins necessary to save the world, to an internal quest of self-realization.
Quests continued in modern literature. Analysis can interpret many (perhaps most) stories as a quest in which the main character is seeking something that he desires, but the literal structure of a journey seeking something is, itself, still common. Quests often appear in fantasy literature.
Adventure travel narratives are often written within a quest genre. The quest as genre is a romantic narrative that follows a pattern of sequential steps: the call to journey, preparation, the journey, and returning home.
[An illustration of the quest genre as spiritual metaphor in adventure travel narratives by Jasmine M. Goodnow & Edward Ruddell http://www.tandfonline.com/ ]
Diagramma poem from Angels & Demons:
From Santi’s earthly tomb with demon’s hole,
‘Cross Rome the mystic elements unfold.
The path of light is laid, the sacred test,
Let angels guide thee on thy lofty quest.
As you said, quest genre sounds like a sub-genre of fantasy because it commonly appears in fantasy novels or movies, but not necessarily. Quest fantasy or fantasy quest is often used as a sub-genre of fantasy to differentiate from the general quest genre.