I tried to get convenient with some techniques to find the index of an ideal. For example we could consider the quotient ring $\mathbb{Z}[\sqrt{-19}]/I$ where $I:=(4,2+2\sqrt{-19})$ and find its cardinality.
One way to start is by mentioning that $$\mathbb{Z}[\sqrt{-19}]\cong \mathbb{Z}[X]/(X^{2}+19).$$ And consequently we see that $$\mathbb{Z}[\sqrt{-19}]/I\cong \mathbb{Z}[X]/(X^{2}+19,4,2+2X),$$ and therefore we see that $$\mathbb{Z}[\sqrt{-19}]/I\cong(\mathbb{Z}/4\mathbb{Z})[X]/(X^{2}-1,2+2X).$$
In this last ring we see that $X^{2}\equiv 1$ and $2X\equiv -2$, so informally I see that we have representatives of the form $a+bX$ where $a\in\{0,1,2,3\}$ and $b\in\{0,1\}$, so that we have $8$ elements. But how do we formally conclude this last claim? Is there in general a more easy strategy to tackle such problems? For instance how could one finish the example above? In advance, thank you for your help!
The general problem of determining the index of an ideal in an order is not easy. There are deterministic algorithms to determine whether the index is finite, and then compute it, but doing this by hand quickly becomes unmanageabale. For 'small' orders, such as orders generated (as an algebra over $\Bbb{Z}$) by one or two elements with minimal polynomials of degree up to $4$ or $5$, it is very doable to determine the index by hand. See below for details.
In your example you are right to immediately note that $$\Bbb{Z}[\sqrt{-19}]/I\cong(\Bbb{Z}/4\Bbb{Z})[X]/(X^2-1,2X+2),$$ which already shows that the index divides $16$, because the quotient is generated by $1$ and $X$ over $\Bbb{Z}/4\Bbb{Z}$ (as an additive group). More precisely, we see that the order of the quotient $$Q:=(\Bbb{Z}/4\Bbb{Z})[X]/(X^2-1),$$ is exactly $16$, and its ideal $(2X+2)Q\subset Q$ is easily verified to contain only $2$ elements. It follows that $$(\Bbb{Z}/4\Bbb{Z})[X]/(X^2-1,2X+2)\cong Q/(2X+2)Q,$$ and so the order of this quotient is $\tfrac{16}{2}=8$.
In general, if you want to determine the index of an ideal $I\subset\Bbb{Z}[\alpha]$, where $\alpha$ is a zero of a monic polynomial $f\in\Bbb{Z}[X]$, you can start by picking some $g\in\Bbb{Z}[X]$ of low degree such that $g(\alpha)\in I$. If you have a constant $c\in I$ as in your example, then immediately $$\Bbb{Z}[\alpha]\cong \Bbb{Z}[X]/((f)+I)\cong(\Bbb{Z}/c\Bbb{Z})[X]/((f)+I).$$ This is a quotient of a finite ring, and if $\deg f$ and $c$ are not too big you can explicitly count the number of elements in the ideal $(f)+I$, and often you can use ad hoc arguments for such a count.
If you do not have a constant $c\in I$, then pick $g\in\Bbb{Z}[X]$ of low degree such that $g(\alpha)\in I$ and compute the resultant $R(f,g)$. If $f$ and $g$ have no nonconstant common factor, then $c:=R(f,g)$ is a constant contained in $I$ and the process above applies.