Are the sets of all finite automata and pushdown automata countable?
The set of all finite automata is indeed countable, but that has nothing to do with the set of all TMs being countable. The two models are only related by the set of languages they accept, but both have infinitely many machines per language so we can't immediately conclude anything about the sets of machines.

You can prove countability with the usual means: working from the formal definition of FAs (something like $A = (Q, \Sigma, \delta, Q_F)$), construct an injection into $\mathbb{N}$. That's tedious, but elementary. For PDAs, it works just in the same way.