Julie,

I anticipate that your idea of a 'net' would be a series of flat pieces of material (paper), with straight sides, which when folded up and taped would form a sphere.

The short is answer is that no finite number of pieces would work.
The 'idea' is that when you fold up a net, all parts except the vertices are flat. That is, for example if you take three points nearby, Pythagoras theorem works, or that if you draw a small triangle, the sum of the angles is 180 degrees. Neither of these is true for even a small (but finite) piece of a sphere.

When computer scientists want to makes something that 'looks like' a sphere, they use a very large number of very small polygons - but if you zoom in it either looks distorted or they recompute with an even larger number of even smaller pieces to keep this 'out of sight'!

The alternative is that you are trying to do something with a rubber sheet, and list faces, vertices and edges. If that is the goal, then you need to switch to 'rubber sheet geometry' which is topology. Then many of the standard rules need to be rethought to keep some larger patterns working - and the answer is a bit long.

Walter Whiteley