Self taught here so please bear with me. How does one define the ring of integers of the field $\mathbb{Q}(r)$, where $r$ is a root of the cubic $$x^3+px+q$$ as well as determining the fundamental units of this field, alongside determining which primes are irreducible and reducible (assuming it's a principal ideal domain).
If it helps, assume I know how to calculate the coefficients of the minimal polynomial to an element of the field with respect to whatever the basis is.
In complete agreement with Mariano Suárez-Álvarez’s comment, I would suggest that if you have a specific case of a cubic, and $\lambda$ is a root, look at the minimal polynomial of $A+B\lambda+C\lambda^2$, and see whether, by guess and by golly, you can figure out the complete criteria for this general element of $\Bbb Q(\lambda)$ to be an algebraic integer. In my experience, this can be the very most difficult part of the task.
Then, believe it or not, finding the class group is comparatively easy. But even when you know (as is the case when the field has one real archimedean absolute value and one (pair of) complex, so that there is only one primitive unit), finding a basis for the units can be extremely difficult with only elementary methods.