The Laplace domain transfer function, if properly derived, is Vout/Vin where Vout and Vin are the Laplace transforms of output and input voltages. You multiply the Laplace transform of the input voltage with the transfer function and find the inverse transform of the result. That's your Vout in time domain.
Pure sinusoidal inputs can be handled in frequency domain, you simply calculate the gain or attenuation and the phase shift. Your transfer function is a multiplying phasor when you calculate it as complex number in the operating frequency.
If you calculate the inverse Laplace transform of the Laplace domain transfer function, you get what the system outputs in case the input is the unit impulse function (=Dirac's delta)
Unfortunately often the inverse transform is too complex. Then you must calculate the output numerically. Circuit analyzing programs do it automatically.