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I want to know how to go about describing a ring after adjoining an element that satisfies a certain relation.

As an example, I'm considering the ring obtained from Z3 by adjoining an element a satisfying $a^2 + a + 1=0$

So we consider $Z3[x]/x^2 + x + 1$

The way I'm thinking about it, is that we want to find all the divisors of $x^2 + x + 1$ first. We know that the polynomial has linear factors since it's reducible. Namely $(x-1)$. We also know all elements in this ring will have deg. 0 or 1. I think we should have elements $0, 1, 2$. But how can I get the rest of the elements in the ring?

Also any general tips on how to approach this type of problem with arbitrary rings/polynomials?

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    I suggest you look up Matthew Salomone and his youtube series "Exploring Abstract Algebra" or something like that. His way of explaining concepts like this appealed to me and you might find it useful too. It might also be useful to watch his videos several times because he explains things quite quickly, but also concisely. – 2017-02-20

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By the division algorithm, any polynomial $f(x) \in (\mathbb{Z}/3\mathbb{Z})[x]$ can be written as $$ f(x) = q(x) (x^2 + x + 1) + r(x) \equiv r(x) \pmod{x^2+x+1} $$ for some $q(x),r(x) \in (\mathbb{Z}/3\mathbb{Z})[x]$, where either $\deg(r(x)) < 2$ or $r(x) = 0$. This shows, as you stated, that every element of $(\mathbb{Z}/3\mathbb{Z})[x]/(x^2+x+1)$ has a representative of degree $0$ or $1$. Such a representative is of the form $a + bx$ with $a,b \in \mathbb{Z}/3\mathbb{Z}$. Does this help you see what the other elements of $(\mathbb{Z}/3\mathbb{Z})[x]/(x^2+x+1)$ look like? How many are there?

On the other hand, the factorization of $x^2 + x + 1$ over $\mathbb{Z}/3\mathbb{Z}$ gives you a different sort of information. We know $(x-1)(x^2+x+1) = x^3 - 1 = (x-1)^3$, so $x^2 + x + 1 = (x-1)^2$. Sending $x$ to $y+1$, we have $$ \frac{(\mathbb{Z}/3\mathbb{Z})[x]}{(x-1)^2} \cong \frac{(\mathbb{Z}/3\mathbb{Z})[y]}{(y^2)} \, . $$ This shows us that the quotient ring is isomorphic to the ring of dual numbers. In particular, it has nontrivial nilpotent elements: $\overline{y}^2 = \overline{y^2} = 0$, but $\overline{y} \neq 0$.