I started by making a truth table for A and B, with
$\begin{array}{c:c|c}A&B&A \wedge B\\\hline T&T& T\\ T&F& F\\ F&T& F\\ F&F& F\end{array}$
To see what would happen, I made a table for ~A and ~B:
$\begin{array}{c:c|c}\neg A&\neg B&\neg A \wedge\neg B\\\hline F&F& T\\ F&T& F\\ T&F& F\\ T&T& F\end{array}$
I am confident this is wrong, it doesn't make sense that the union of two statements is equal to the union of the opposite of those statements, but as far as I know, it also does make sense that the output for the first line is true if I say "neither statement A nor statement B is true" as it satisfies the condition of the union of the negation of the original statements.
Why is the table wrong? Why doesn't the union of those negations equal the preceding table?
They are not equal. Note that the second table is upside down (relative to the first).
Put them together and you get:
$$\begin{array}{c:c|c:c|c:c} A & B&\neg A& \neg B& A\land B& \neg A\land\neg B\\\hline T & T &F&F& T & F\\ T & F&F&T & F & F\\F & T&T&F& F& F\\F & F& T&T&F & T\end{array}$$