How do I prove that $x^TAy = y^TAx$ if A is symmetric?

$x^tAy$ is a scalar. So $x^tAy=(x^tAy)^t$. Can you continue from here?

If $A$ is symmetric then we know that $A_{ij} = A_{ji}$. If you understand that $x^T A y$ = $\sum_i\sum_j x_iA_{ij}y_j$ , then swapping the indices of $A$ should directly lead you to the answer.

Prove that whenever $A$ and $B$ are matrices for which you can compute the product $AB$, then $$(AB)^t=B^tA^t$$.

Next apply $(\mathord-)^t$ to the left hand side of your equation, and compare the result to the right hand side, keeping in mind that both sides are actually numbers (well, $1$-by-$1$ matrices)