There are no perfect answers.
A simple technique is like the skybox - a cube of 6 square textures that get radially projected down onto the sphere.
Equal area projections can be good to ensure "pixels of texture per unit area of sphere" is constant. But there can be unfavourable stretching in one direction and squashing in another direction.
Equidistant projections kind of do something opposite, and ensure there is no local stretching and squashing, but don't guarantee pixels:area constancy.
One projection is both equal area and is equidistant (but in a restricted way): Lambert azimuthal equal-area projection. You may find this works well