Let $H$ be a subgroup of $G$ and $\DeclareMathOperator{\GL}{GL} \rho: G \rightarrow \GL(V)$ a representation of G. The restriction map $\rho |_H$ is a group homomorphism, i.e., a representation of $H$.
I have calculated the character tables for $S_4$ and $A_4$:
$S_4$: \begin{array}{|*{3}{r|}} {} & e & (12) & (12)(34) & (123) & (1234) \\ \hline \text{Size}& {1} & {6} & 3 & 8 & 6 \\ χ_1 & 1 & 1 & 1 & 1 & 1 \\ χ_1 & {1} & {-1} & 1 & 1 & -1 \\ χ_2 & {3} & {1} & -1 & 0 & -1 \\ χ_3 & {3} & {-1} & -1 & 0 & 1 \\ χ_4 & {2} & {0} & 2 & -1 & 0 \\ \hline \end{array}
$A_4$: \begin{array}{|*{3}{r|}} {} & e & (12)(34) & (123) & (132) \\ \hline \text{Size}& {1} & {3} & 4 & 4 \\ χ_1 & {1} & {1} & 1 & 1 \\ χ_2 & {1} & {1} & \omega & \omega^2 \\ χ_3 & {1} & {1} & \omega & \omega^2 \\ χ_4 & {3} & {-1} & 0 & 0 \\ \hline \end{array}
Now for each irreducible representation of $S_4$, determine whether the restriction on $A_4$ is an irreducible representation of $A_4$, and if not write it as a direct sum of irreducible ones. Any help would be appreciated!
First, there are a couple of typos in your tables: for $S_4$ you've labeled both the trivial representation and the sign representation by $\chi_1$. And more seriously, for $A_4$ you have identical rows for $\chi_2$ and $\chi_3$.
Let $\rho$ be a representation of $S_4$ with character $\chi$. Since conjugacy classes of $S_4$ are determined by cycle type, then $\chi$ takes the same value on any two permutations with the same cycle type. Restricting doesn't change the value of the representation, so this also true for $\chi|_{A_4}$. In particular, this means that $\chi|_{A_4}(1\ 2\ 3) = \chi|_{A_4}(1 \ 3\ 2)$.
Here's an example to help you compute the characters of the restricted representations. Let $\rho$ be the irreducible representation of $S_4$ whose character you've labeled $\chi_2$, and let $\psi$ be the character of $\rho|_{A_4}$. Since transpositions and $4$-cycles do not belong to $A_4$, we can ignore these columns. For the rest, we simply copy the values of the row for $\chi_2$, obtaining the following row. $$ \begin{array}{c||c|c|c|c} A_4 & e & (1\ 2)(3\ 4) & (1\ 2\ 3) & (1\ 3\ 2)\\ \hline \psi & 3 & -1 & 0 & 0 \end{array} $$ These are exactly the values of $\chi_4$ (in your $A_4$ table), so $\psi = \chi_4$ and $\rho|_{A_4}$ is irreducible. Try to fill in the rest of the values of the table yourself.
As we saw in the example, it's easy to tell if the restricted representation is irreducible: you've listed all the irreducible characters of $A_4$ so if the restriction is irreducible, its character will appear there. For those restrictions that are reducible, we can express them as sums of irreducible characters using the inner product. Since $\langle \eta, \chi_j \rangle$ is the multiplicity of $\chi_j$ in $\eta$, then \begin{align*} \eta = \sum_{j=1}^4 \langle \eta, \chi_j \rangle \chi_j \, . \end{align*}