Specifically what does stuff like $\mathbb{Q}[x] / \langle x^{2} - 1\rangle$ generally means? I thought it meant the set of polynomials with rational coefficients divided by $x^{2} - 1$, but when I see stuff like $\mathbb{Z}[x]/\langle x^{2} + 1, 5\rangle$ I have no idea what it means.
Also, how do you generally find what $R/I$ is isomorphic to?
Usually, you'd learn that quotient rings are sets of certain residue classes with a certain ring structure imposed. But this really obscures the vision on what quotient rings really are: If $R$ is a commutative ring and $A\subseteq R$ some subset, then conceptually speaking, $R/\langle A\rangle$ is a ring with essentially the same structure as $R$, but with the additional assumption that every element of $A$ is equal to $0$. Of course, if $a_1$ and $a_2$ are $0$, their sum should also be $0$. And if $a$ is $0$ and $r$ is some other element of $R$, then their product should also be $0$. So if every element of $A$ is $0$, then all the elements of the ideal $\langle A\rangle$ should also be $0$, since those elements are exactly those which can be generated by the mentioned operations (adding elements toeach other or multiplying them by some random element). So essentially, $R/I$ with some ideal $I\subseteq R$ is a ring with the structure of $R$, but the additional assumption that every element of $I$ is $0$, and no other elements are $0$.
We can also deduce more relations from this fact. For instance, if $a-b$ is in $I$, then $a-b$ is $0$, so $a$ and $b$ are equal. Such relations must hold in the ring $R/I$ if elements of $I$ are supposed to be $0$. But these forced relations are the only relations that differentiate $R$ from $R/I$. Every single rule about how to do algebra in $R/I$ follows from the two rules that 1. everything works like in $R$, except for the fact that 2. if something is in $I$, it's $0$. You don't need to know more than this to do algebra in $R/I$.
For instance, consider $\mathbb Q[X]/\langle X^2+1\rangle$. I will add a prime to elements of the quotient to differentiate them from elements of the additional ring. At first, this quotient ring is made up of elements which look like those of $\mathbb Q[X]$, so polynomials of the form $a_0'+a_1'X'+\dots+a_n'X'^n$. But we get the additional rule that $X'^2+1'=0'$, or in other words, $X'^2=-1'$, from which we can deduce $X'^{2k}=1'$ if $k$ is even, and $X'^{2k}=-1'$ if $k$ is odd. So those polynomials reduce to $$a_0'+\dots+a_n'X'^n=(a_0'-a_2'+a_4'-\dots) + (a_1'-a_3'+a_5'-\dots)X'.$$ Those sums can be replaced by their results, say $b'$ and $c'$, to make them into $$b'+c'X',$$ where $b'$ and $c'$ behave like $b$ and $c$, and $X'^2=-1'$. So it behaves exactly like $a+b\mathrm i$ should in $\mathbb Q[\mathrm i]$, the ring of rationals with $\mathrm i$ added. In fact, it's a nice way to define $\mathbb Q[\mathrm i]$.
Now as an afterthought, we might want to guarantee that such a quotient ring actually exists for all rings and all their ideals. We do that by constructing them explicitly using residue classes. But residue classes are not what quotient rings are about, it's just a tool to guarantee their existence.
