Actually, we have a very good description about the nature of gravity on all but the quantum level. It's called the Theory of General Relativity. Speaking strictly of the classical Newtonian Theory (which I believe you are), it's a moot point whether it's a 'push' or a 'pull' so long as Newton's 3rd Law holds. It's a force, and that's all that matters - the sign ambiguity cancels.
>>it's a moot point whether it's a 'push' or a 'pull' so long as Newton's 3rd Law holds.<<
True, if one is not curious. But KNOWING which it is could greatly impact our understanding of the physical universe and how it works. And the more we know about it, the more we can manipulate it.