Difference between 游学 and 留学
留学 http://www.cantonese.sheik.co.uk/dictionary/words/20426/
to study abroad

'留' = 'stay'; '學' = 'study'
'留学' means 'stay (abroad) to study'
If you say: &quot;我要去留學&quot; (I am going to study abroad), it implies there is a clear destination. e.g. &quot;我要去美国留学&quot; (I am going to America to study). In this case, you are going to enroll to a particular college or university in a particular city in a foreign country
'游学' is short for 一邊游历一邊學習 (learning while traveling). It is a very general term.
'游学' could mean you are traveling across your own country and learn new things as you go, from serious study like taking college courses, to casual learning like learn to make wonton noodle soup from a restaurant owner; It could also mean you go to many foreign countries and studying in many different universities or taking apprenticeship under different masters.
'四方游学' is a common phrase that describe 'go to many places to study' or 'traveling the world and study along the way'   . But there's no such thing as '四方留学'