This question is very basic, but I couldn't find a definitive answer.
I know that if two matrices are similar, they have the same characteristic polynomial. However, if two matrices don't have the same characteristic polynomial, are they guaranteed to be not similar?
Yes. The contrapositive of this says if two matrices are similar, then they have the same characteristic polynomial. Suppose $A$ and $B$ are similar. Then $A=P^{-1}BP$ for some invertible matrix $P$. Then
$$\chi_A(t)=\det(tI-A)=\det(tI-P^{-1}BP)=\det(P^{-1}(tI-B)P)=\det(tI-B)=\chi_B(t)$$
because the determinant is multiplicative.