A is a finite subset of S, which is an infinite set.
How can I prove that $|S| = |S \setminus A|$?
I just finished proving that $|T \cup S|$ where $T$ is infinite and $S$ is countable is $|T|$. They seem related but I can't solve it because I am confused by the concepts.
Can someone help me?
Given that you proved that $|T\cup S| = |T|$ if $|T|$ is infinite and $|S|$ is countable, you can consider the identity $$S = (S\setminus A)\cup A.$$ Here, $A$ is countable (because it's finite), and $S\setminus A$ is infinite (because if it were finite then so would be $S$). By your identity, $$|S| = |S\setminus A|.$$
In general, if $X$ and $Y$ are disjoint then $|X \cup Y| = |X| + |Y|$. When not both summands are finite, cardinal addition breaks down into taking maxes: $$|X| + |Y| = \max\{|X|, |Y|\}.$$ In your situation, let $X = S\setminus A$ and let $Y = A$.