1
general::
Make progress, succeed, as in I suspect they'll be going places with the new product, or Now that she has her doctorate I'm sure she'll go places. [Colloquial; early 1900s]
American Heritage Idioms
2
general::
Phrase(s): go places
to become very successful. • I knew that Sally, with all her talent, would go places. • I really want to go places in life.
McGrawhill's American Idioms And Phrasal Verbs