First I distributed the $\bar A$ and got $\bar{A}BA+\bar{A}\overline{AB}C$. Then I thought that the $A$ and $\bar{A}$ would cancel and get $B0 + \bar{A}\overline{AB}C$. I then broke up the $A$ and $B$ that were under the line together to get $0 + A(\bar{A}+\bar{B})C$. Then I figured the $A$'s would cancel, and I would be left with $\bar{B}C$.
I am not sure if I'm doing this right, could someone let me know if I messed up, and if so where?
You started out fine, but you lost an overline on the initial $A$ when you expanded $\overline{AB}$ to $\bar{A}+\bar{B}$, so the rest is wrong. Here’s the complete calculation:
$$\begin{align*} \bar{A}(\bar{B}A+\overline{AB}C)&=\bar{A}\bar{B}A+\bar{A}\overline{AB}C\\ &=B\cdot0+\bar{A}(\bar{A}+\bar{B})C\\ &=0+\bar{A}\bar{A}C+\bar{A}\bar{B}C\\ &=\bar{A}C+\bar{A}\bar{B}C\\ &=\bar{A}\cdot1\cdot C+\bar{A}\bar{B}C\\ &=\bar{A}(1+\bar{B})C\\ &=\bar{A}\cdot 1\cdot C\\ &=\bar{A}C \end{align*}$$