What is $|P( P (\{ 1 \})) | $? I recently learned about power sets and I have no idea how to really answer this question.
Here's my attempt:
 $$|P( P (\{ 1 \})) | = |P(\{ \emptyset ,  \{ 1 \} \}) | = |\{ \{\emptyset\},\{\{1\}\}, \{\emptyset, \{ 1\}\}, \emptyset\}| = 4$$

Your expansion is correct, but the question asks for the number of elements in your set, so the answer is 4. Another way to solve this: if $S$ is a set with finite cardinality $|S|=c$, then $|P(S)|=2^c$. So $|P(P(\{1\}))| = 2^{2^1} = 4$.
