What are the morphisms and objects we must assume that are actually there? I'm reading Awodey's: Category Theory. I harbor a little confusion: When we speak about a category, say: $\mathbf{Set}$. In the book, usually he talks about this category but there is no notion of quantity of sets. How many sets are actually there?

Another thing that is getting me confused is this: Suppose that I have this category with $n$ sets, I must have the arrows, the composition, the identity, etc. But for these sets, should I assume that functions can be constructed from one set to every other set? That is: For a category with $3$ objects: $A,B,C$, should I assume that I have the following functions?

$$f_1:A\to B\\f_2:A\to C\\f_3:B\to C\\f_4:C\to B\\f_5:C\to A\\f_6:B\to A\\$$

I'm not sure if it's about what I have or what could be made.
First, "Set" is one specific category. Its objects are all the sets and its arrows are all the functions between sets. 

In the second paragraph, you suppose you have a category with n sets. To define the category you have to say specifically which n sets the objects are.  You then have to say what the arrows are (they will be set functions in this case). Your specification of what the arrows are must satisfy the axioms for a category: (1) If a set S is an object, the identity function on S must be an arrow.  (2) If you have an arrow f:S\to T and an arrow g:T\to U then you must specify that the composite g\o f is also in the category.  If you do those things, then associativity (in this case) will be automatically satisfied, since composition of set functions is associative. Once you have done those things you have defined a category.

In your second paragraph you mention functions f_1:A to B and so on, but you didn't say what the functions are. Also, you mentioned f_1:A to B and f_3:B to C but you didn't mention any function from A to C, which is required by the axiom that says composites exist. 

Also, you ask, "Can I assume that arrows can be constructed from one set to every other set?" That is wrong in two ways: (1) A category with two objects S and T is not required to have an arrow from S to T. (2) You can't "assume" there are arrows, you have to say specifically what they are.
How many sets are actually there?


The powerset of all sets P(SET) in the reals is infinite. Hence, from an infinite number of real numbers you can create an infinite number of sets.  This is not true of all powersets P(_), but true of SET.


  Suppose that I have this category with nn sets, I must have the arrows, the composition, the identity, etc. But for these sets, should I assume that functions can be constructed from one set to every other set?


P(n), the power set of all sets created up to "n" will give you all possible combinations.  Remember, SET doesn't have arrows (no structure beyond associativity and commutivity of elements) except from powerset to powerset.


  That is: For a category with 3 objects: A,B,CA,B,C, should I assume that I have the following functions?


Absolutely not; you should never "assume" any functions in CT. The only categorical assumption you are allowed of sorts is that if you assume you are dealing with a category, then you can assume that composition and associativity apply. 

Instead, the powerset of 3 objects is:

$$P(3) = (a,b,c), (b,c,a), (c,a,b), (b,a,c), (a,c,b), (c,b,a), (a,c,b), (c,b,a), (b,a,c)$$

Thus the only arrows possible are those that take you from one set to the other:
eg. between (a,b,c) and (b,c,a) you have the following:

$$ f1:a \rightarrow b$$ 

$$f2: b\rightarrow c$$

$$f3: c\rightarrow a $$