Entropy and enthalpy compensation are frequently observed in thermodynamic measurements. To understand the molecular details of these compensating changes, molecular dynamics or Monte Carlo methods may, in principle, be used. Yet, in practice, computational methods to evaluate changes in entropy and enthalpy usually have much greater (e.g. order of magnitude worse) error than relative free energy calculations. In this paper, we examine ways to improve the computational ability to determine changes in both enthalpy and entropy. Toward that end we consider five different perturbation schemes for enthalpy determination and also consider the relative importance of staging the computation (i.e., dividing the entire perturbation calculation into a series of consecutive smaller ones) along the reaction coordinate. Two model systems are used for evaluation: a system of Lennard-Jones spheres, where the computation is directed to the addition/deletion of a sphere, and the alchemical transformation of an anion in explicit TIP3P water from one type to another. The results show only subtle differences among the better perturbation methods and much larger improvement in relative accuracy with the use and careful selection of staged intermediates along the reaction coordinate. As a result of these calculations, we suggest that approximate calculations of relative entropy be used to guide the staging of intermediates for computation of free energy, entropy, and enthalpy differences. We suggest that for these purposes the single state perturbation (SSP) method is the best choice. For the pure purpose of enthalpy and/or entropy evaluation, we suggest multistage SSP and direct approach, respectively, for systems with small and large differences.