What does it mean to "prove $1+1=2$"? It is a famous bit of trivia that it took Russell and Whitehead about 300 pages to prove that $1+1 = 2$. However, this seems more like a definition rather than theorem. As far as I know, $2$ is just the symbol we use as shorthand for $1+1$, where $1$ is the multiplicative identity.
You have to define what system you are working in.  In Peano Arithmetic, PA, $1$ and $2$ are not part of the language.  They are abbreviations for $S(0)$ and $S(S(0))$, where $S$ is intended as the successor function, so you are being asked to prove $S(0)+S(0)=S(S(0))$.  You can follow the Wikipedia proof, which you may need to update a bit depending on how your version of the axioms is written.  Basically this should look a lot like your axioms that define addition.  PA does not specify a multiplicative identity, you have to prove that $S(0)$ is one by induction.
Axioms for the natural numbers usually define only a single number: either $0$ or $1$. The existence of any other natural number must be inferred from these axioms using the axioms and rules of logic or set theory (if applicable). If addition on $N$ is given in your axioms, it would be a trivial exercise to prove that 1+1=2. 

Suppose, for example, you are given the axioms: 


$1\in N$
$\forall x \in N: \exists y\in N: y=x+1$


Applying both of these axioms, we can obtain:

$\exists y\in N: y=1+1$

Applying the rule of existential specification (for $y = 2$), we could infer that $2 \in N$ and $2=1+1$. This inference can be thought of as the "definition" of $2$. 

If addition on $N$ is not given in your axioms, you would have to construct (i.e. prove the existence of such a function) using the axioms of your set theory before you could "define" or infer that $2=1+1$ as above.