A comprehensive study of integral equation theories for binary mixtures is presented. The mixture components differ primarily in size (the diameters differ by 10%, 20%, and 30%) and interact either via hard potentials or via Lennard-Jones potentials. For the latter, variation with energy parameter (epsilon) is considered. This article focuses on improving the accuracy of the theories by systematic inclusion of bridge diagrams into the closure relationships. Specifically, the first two orders of bridge diagrams are exactly evaluated for these mixtures. A general Monte Carlo integration scheme for diagram evaluation is discussed and applied. Comparisons with diagrams obtained from a Legendre expansion technique are made in order to assess whether this approach is practical. The approximation of higher order diagrams has been considered. Specifically, techniques for approximation of all higher order diagrams, which were successful for single component fluids, were found to be problematic for mixtures. However, a simple algorithm for approximate third order diagrams is presented and found to lead to improvements. A detailed analysis of the bridge diagram variation with the nature of the mixture is presented and may be useful in extending the present results to related mixtures. The spatial dependence of the diagrams has also been examined and found to be extremely well reproduced by simple polynomial expansions. In addition, physical arguments have been applied to extract large separation limits of the diagrams. The accuracy of the integral equation theories with order of bridge diagrams is assessed by comparing pressure estimates from the virial expansion and from the integration of compressibilities. With this measure, the quality of the integral equation theories for each mixture is assessed at 18 state points. In all cases, the thermodynamic consistency improves smoothly and rapidly with the order of bridge diagram included in the theory. This result, together with the general Monte Carlo algorithm and the detailed structural and spatial analysis, shows that direct bridge diagram evaluation is practical and consistently improves the quality of the theory for these mixtures.