Meteorites provide an opportunity to reconstruct the history of the SolarSystem. Differentiated meteorites, also called achondrites, are the result of melting and differentiation processes on their parent body. Stable isotopic compositions of differentiated meteorites and their components have added to the understanding of physical parameters, such as temperature, pressure, and redox conditions relevant to differentiation processes on planetesimals and planets in the early Solar System. In particular, Fe and Si isotopes have proven to be useful in advancing the understanding of physical and chemical processes during planetary accretion and subsequent evolution. In this work, I developed a new method to simultaneously purify Fe and Si from a single aliquot of sample while ensuring consistently high yields and accurate and precise isotopic measurements. I then measured the Fe isotope compositions and Si contents of metals from aubrite meteorites to infer the structure and thermal evolution of their asteroidal parent body. Thereafter, I determined the combined Si and Fe isotope compositions of aubrite metals and the Horse Creek iron meteorite, and compared the magnitude of Si and Fe isotope fractionation factors between metal and silicates for both enstatite chondrites and aubrites to estimate the effect of high-temperature core formation that occurred on the aubrite parent body. I additionally assessed whether correlated Si and Fe isotope systematics can be used to trace core formation and partial melting processes for the aubrite parent body, angrite parent body, Mars, Vesta, Moon, and Earth. Finally, I measured the combined Fe and Si isotope composition of a variety of ungrouped achondrites and brachinites that record different degrees of differentiation under different redox conditions to evaluate the role of differentiation and oxygen fugacity in controlling their Fe and Si isotope compositions. Taken together, this comprehensive dataset reveals the thermal evolution of the aubrite parent body, provides insights into the factors controlling the Fe and Si isotope compositions of various planetary materials, and helps constrain the bulk starting composition of planets and planetesimals.