I would like to diagram the complete lattice of ideals of $R = \mathbb Z[x]$ containing the ideal $I = (3, x^3 - 1)$.
By the lattice isomorphism theorem, each ideal of $R$ containing $I$ corresponds to an ideal of $R/I$. Now $R/I \cong (\mathbb Z/3\mathbb Z)[x]/(x^3 - 1) = \{(ax^2 + bx + c) + I \mid a, b, c \in \mathbb Z/3\mathbb Z\}$ has 27 elements.
What is the fastest way to identify ideals in $R/I$ and the corresponding ideals in $R$?
Good start! To finish, you should just realize that $\Bbb Z/3\Bbb Z$ is the field of three elements, so you are looking at a quotient of $F_3[x]$, which is a principal ideal domain.
Each of the ideals is generated by a unique monic polynomial, and ideal containment is reflected by divisibility. So, the ideals between $(x^3-1)$ and $F_3[x]$ correspond to divisors of $x^3-1$ over $F_3$. Each of those ideals corresponds to an ideal between $(3,x^3-1)$ and $\Bbb Z[x]$.