How can I change the color of the (anchor handle and path) lines that illustrator shows between two points?
In my case, the line is black and it's impossible to see if I'm working with a black background.
Those lines will be whatever color you have selected for your layer.
If your layer color is black, then those will also be black.
Just change the color of your layer by double clicking on the layer and selecting a different color.
To open the Layers Panel, go to Window → Layer or hit F7
There is a list of colors to choose from, but if you'd like you can also select custom and choose any color you want.