How can I make it so that the gradient classifer (look closely, there are bits of red in some places) is more pronouced? Or that the outlines of the zip codes are less pronounced?
Go to Layer Properties > Style. I like to use Rule-based styling which you can select from the drop-down menu and create how each attribute or column should be visualised. If you double-click on the label to access Rule Properties, you can edit how lines should be stylised such as its width, colour, angles etc.
I used "Zip codes" as a filter (you can change this to suit your column name) and you edit the Outline width to make all polygons with a value in the zip code column have a lighter/darker outline if you wish. You can also edit the filter to use specific styles for specific zip codes (eg. "Zip codes" = 'T67 H7').
Hope this helps.