Electron-electron interactions between two closely spaced electron gases can be directly measured by the drag voltage induced in one layer when a current is driven through the other. We have used such drag measurements of GaAs/AlGaAs double quantum wells to probe both the electronic excitations within each layer and the interlayer Coulomb interaction. Two contrasting regimes have been explored. First, at high temperatures we measure an enhanced drag due to the "antiscreening" of the coupled plasmon modes - a result which is in good agreement with theoretical predictions. Moreover, we show the importance of many-body corrections to the plasmon modes, enabling us to test a local-field description of the correlations. Second, drag measurements in the quantum Hall regime show more pronounced structure than the Shubnikov-de Haas oscillations; the drag is more sensitive to spin splitting at high filling factors and enhances the features seen at low filling factors. We show that the localised states away from the Landau level centre influence the drag.