I have taken a course on schemes, so I am familiar with the basic definitions, but I'm very rusty and I've forgotten how to do this (if I ever knew).
Basically, I want to compute the residue fields of points on affine schemes like $$ X = \operatorname{Spec}\Big(\frac{\mathbb{F}_p[x, y]}{(x^2 + y^2)}\Big). $$ For example, I'd like to compute the residue field of $X$ at the point $(0,0)$, by which I mean the prime ideal $(x, y)$. Algebraically, I want to compute the residue field of the local ring $$ \Big( \frac{\mathbb{F}_p[x,y]}{(x^2 + y^2)}\Big)_\mathfrak{m}, $$ where $\mathfrak{m}$ is the (maximal) ideal generated by $x$ and $y$. This seems like a bit of a nightmare to do by hand, but I suspect there's some geometric machinery that can deal with it quite easily.
A useful fact here is that if $A$ is a commutative ring and $P\subset A$ is a prime ideal, then the residue field of the localization $A_P$ is naturally isomorphic to the field of fractions of the quotient $A/P$. (Basically, to get the residue field, you need to kill $P$ and invert each element of $A\setminus P$, and it doesn't matter which order you do these two steps in.) In the case of a maximal ideal, the residue field of the localization is just the same as the quotient $A/P$. So in your case, instead of having to worry about what your localization looks like, you can just mod $\mathbb{F}_p[x,y]/(x^2+y^2)$ out by the ideal $(x,y)$, which just gives you $\mathbb{F}_p[x,y]/(x,y)\cong \mathbb{F}_p$.