We study approximations of modules of finite Gorenstein flat dimension by (projectively coresolved) Gorenstein flat modules and modules of finite flat dimension. These approximations determine the Gorenstein flat dimension and lead to descriptions of the corresponding relative homological dimensions, for such modules, in more classical terms. We also describe two hereditary Hovey triples on the category of modules of finite Gorenstein flat dimension, whose associated exact structures have homotopy categories equivalent to the stable category of projectively coresolved Gorenstein flat modules and the stable category of cotorsion Gorenstein flat modules, respectively.