Strongly interacting Fermi gases provide a clean and controllable laboratory system for modeling strong interparticle interactions between fermions in nature, from high temperature superconductors to neutron matter and quark-gluon plasmas. Model-independent thermodynamic measurements, which do not require theoretical models for calibrations, are very important for exploring this important system experimentally, as they enable direct tests of predictions based on the best current non-perturbative many-body theories. At Duke University, we use all-optical methods to produce a strongly interacting Fermi gas of spin-1/2-up and spin-1/2-down ⁶Li atoms that is magnetically tuned near a collisional (Feshbach) resonance. We conduct a series of measurements on the thermodynamic properties of this unique quantum gas, including the energy E, entropy S, and sound velocity  c. Our model-independent measurements of E and S enable a precision study of the finite temperature thermodynamics. The E(S) data are directly compared to several recent predictions. The temperature in both the superfluid and normal fluid regime is obtained from the fundamental thermodynamic relation T=∂ E/∂ S by parameterizing the E(S) data using two different power laws that are joined with continuous E and T at a certain entropy S _c, where the fit is optimized. We observe a significant change in the scaling of E with S above and below S _c. Taking the fitted value of S _c as an estimate of the critical entropy for a superfluid-normal fluid phase transition in the strongly interacting Fermi gas, we estimate the critical parameters. Our E(S) data are also used to experimentally calibrate the endpoint temperatures obtained for adiabatic sweeps of the magnetic field between the ideal and strongly interacting regimes. This enables the first experimental calibration of the temperature scale used in experiments on fermionic pair condensation, where the ideal Fermi gas temperature is measured before sweeping the magnetic field to the strongly interacting regime. Our calibration shows that the ideal gas temperature measured for the onset of pair condensation corresponds closely to the critical temperature T _c estimated in the strongly interacting regime from the fits to our E(S) data. We also calibrate the empirical temperature employed in studies of the heat capacity and obtain nearly the same T _c. We determine the ground state energy by three different methods, using sound velocity measurements, by extrapolating E(S) to S=0 and by measuring the ratio of the cloud sizes in the strongly and weakly interacting regimes. The results are in very good agreement with recent predictions. Finally, using universal thermodynamic relations, we estimate the chemical potential and heat capacity of the trapped gas from the E(S) data.