For the first question my idea was to show that $\sigma(f(\mathcal{A})) \subseteq \sigma(\mathcal{A})$ and $\sigma(\mathcal{A}) \subseteq \sigma(f(\mathcal{A}))$. As for the second question I am at a loss of what to do there. I have been tinkering with it for awhile now and have not got anywhere. Help on both questions would be much appreciated!
It is clear that $\sigma(\mathcal A)\subset\sigma(f(\mathcal A))$, as $\mathcal A\subset f(\mathcal A)$. For the reverse inclusion, $\sigma(A)$ is a $\sigma$-algebra that contains $f(\mathcal A)$, so as $\sigma(f(\mathcal A))$ is the intersection of all such $\sigma$-algebras, $\sigma(f(\mathcal A))\subset\sigma(A)$.
For the second question, there is a clever argument given in this answer by @Robert Israel: Algebra generated by countable family of sets is countable?
