Proof for Cauchy-Schwarz inequality for Trace

The Cauchy-Schwarz inequality is valid for any inner product, so you just need to show $\operatorname{\textbf{Tr}}A'B$ is an inner product. It's clearly bilinear (or sesquilinear if by $'$ you meant a complex adjoint), with $$\operatorname{\textbf{Tr}}A'A=\sum_i (A'A)_{i}=\sum_{ij}A'_{ij}A_{ji}.$$Depending on whether you're working with the real or complex case, this quantity is either $\sum_{ij}A_{ji}^2$ or $\sum_{ij}|A_{ji}|^2$. Either way it's non-negative, completing the proof.

$Tr(A-tB)'(A-tB) \geq 0$ for all $t$ real. Expand this, use the fact that $Tr(M')=Tr(M)$ (so $Tr(A'B)=Tr(B'A))$ and minimize the left side over $t$. You will get the inequality you want. (This is the standard proof of C-S inequality)