As I understand it,in GAP the function $CharacterDegrees(G)$ gives the degree of irreducible representation the finite group $G$ as pairs $[ m, n ]$ where $m$ is is a degree of irreducible representation, and $n$ is the number of irreducible representations of that degree.
gap> StructureDescription(G);
"C3 x C3"
gap> CharacterDegrees(G);
[ [ 1, 9 ] ]
Does this mean there are nine irreducible representations of $C_3\times C_3$ in $GL_1(\mathbb{C})?$
How do I find the irreducible representations over the reals?
Additionally, is there a way to use GAP to calculate the minimum degree of faithful representations of a finite group?
Yes, there are 9 irreducible representations of $C_3 \times C_3$ over GL$_1(\mathbb{C})$. Finding these 1D representations happens to be relatively straightforward because GL$_1(\mathbb{C})$ is just $\mathbb{C}$, so you are looking for ALL homomorphisms of $C_3 \times C_3$ into the multiplicative group of complex numbers $\mathbb{C}^\times$.
In fact, ANY irreducible representation of an abelian group is a homomorphism of the abelian group into $\mathbb{C}^\times$, and the number of these homomorphisms is exactly the cardinality of the abelian group. This will be helpful for your second question of determining the real representations. A real representation of this group will either have image $\{-1,1\}$ or $\{1\}$ in $\mathbb{C}^\times$. A group with odd order cant have a homomorphic image of even order by the First Isomorphism Theorem, so the only real representation is the trivial representation.
As for your third question, I am not quite sure if such a function exists.