If you have something of the form (y^m)e^(ny) which you need to integrate by parts, you want to differentiate y^m and integrate e^(ny), because this reduces the exponent on the y in subsequent integrals. Note you concluded with "another integration by parts" because you made the problem more difficult.
Also, be flexible. While this is easy to integrate by parts, note that differentiating y*e^-y is going to get you a y*e^-y term back (except maybe with a minus in front). In fact, letting f(y) = -y*e^-y, we see that f'(y) = y*e^-y - e^-y, so that adding Ce^-y to f(y) originally would have given you back y*e^-y upon differentiating (for an appropriate C, whose value should be obvious). Then f(y) + Ce^-y would be your antiderivative.