I've got a mental block on parametrising an ellipse in $\mathbb{A^2}$ over a projective line.
The way I see doing it is this - if you have $$ \frac{x^2}{a^2} + \frac{y^2}{b^2} = c^2 $$ then we can make an affine change of co-ordinates $y' = y-cb$ to put a point of the ellipse at the origin and then you equate points $(s:t)$ in the projective line with points where the lines $sy = tx$ meet the ellipse to get rational polynomial expressions for $x$ and $y$ in terms of $s$ and $t$.
Is this how it works or am I being naive about things? Is there some slicker approach? Later on I'm asked to do the same for a sphere over the complex projective line, but I suppose that is the same deal. I know this seems pretty basic but my head just won't grasp it.
That is the standard approach. You can think of identifying slopes of lines with points in $\mathbb{P}^1$ to get the correspondence.
Alternatively, a maybe more geometric interpretation of this construction as projecting away from the chosen point of the ellipse onto a chosen line — in this case, the chosen line is the line at infinity. You could use any other line for this purpose as well.
Each line through the chosen point will meet the ellipse at a unique other point and the chosen line at a unique point, and conversely either of those points determine the line, so this establishes a one-to-one correspondence between points of the ellipse and points of the chosen line.
(note that the tangent line to the ellipse counts as intersecting the ellipse twice at that point)