Why do we look at morphisms? I am reading some lecture notes and in one paragraph there is the following motivation: "The best way to study spaces with a structure is usually to look at the maps between them preserving structure (linear maps, continuous maps differentiable maps). An important special case is usually the functions to the ground field." 
Why is it a good idea to study a space with structure by looking at  maps that preserve this structure? It seems to me as if one achieves not much by going from one "copy" of a structured space to another copy.  

There is no short and simple answer, as has already been mentioned in the comments. It is a general change of perspective that has happened during the 20th century. I think if you had asked a mathematician around 1900 what math is all about, he/she would have said: "There are equations that we have to solve" (linear or polynomial equations, differential and integral equations etc.).
Then around 1950 you would have met more and more people saying "there are spaces with a certain structure and maps betweeen them". And today more and more people would add "...which together are called categories". 
It's essentially a shift towards a higher abstraction, towards studying Banach spaces instead of bunches of concrete spaces that happen to have an isomorphic Banach space structure, or studying an abstract group instead of a bunch of isomorphic representations etc.
I'm certain all of this will become clearer after a few years of study.
