Understanding why we have these four different energies is crucial to saving yourself time doing thermodynamics problems. In a given problem, identify the variables that are remaining constant. Then choose the energy that has those variables in the differential, so that when you are to calculate the energy, you aren't left with many non-zero terms.
For example, in a process where the volume and number remain constant but the entropy is changing, we should use U, because the Thermodynamic Identity simplifies to dU = τ dσ. Becoming adept at these choices will be critical to your success at solving problems quickly.