The presence of a number of extinct radionuclides in the early Solar System (SS) is known from geochemical and isotopic studies of meteorites and their components. The half-lives of these isotopes are short relative to the age of the SS, such that they have now decayed to undetectable levels. They can be inferred to exist in the early SS from the presence of their daughter nuclides in meteoritic materials that formed while they were still extant. The extinct radionuclides are particularly useful as fine-scale chronometers for events in the early SS. They can also be used to help constrain the astrophysical setting of the formation of the SS because their short half-lives and unique formation environments yield information about the sources and timing of delivery of material to the protoplanetary disk. Some extinct radionuclides are considered evidence that the Sun interacted with a massive star (supernova) early in its history. The abundance of 60Fe in the early SS is particularly useful for constraining the astrophysical environment of the Sun's formation because, if present in sufficient abundance, its only likely source is injection from a nearby supernova. The initial SS abundance of 60Fe is poorly constrained at the present time, with estimates varying by 1-2 orders of magnitude. I have determined the 60Fe-60Ni isotope systematics of ancient, well-preserved meteorites using high-precision mass spectrometry to better constrain the initial SS abundance of 60Fe. I find identical estimates of the initial 60Fe abundance from both differentiated basaltic meteorites and from components of primitive chondrites formed in the Solar nebula, which suggest a lower 60Fe abundance than other recent estimates. With recent improved meteorite collection efforts there are more rare ungrouped meteorites being found that hold interesting clues to the origin and evolution of early SS objects. I use the 26Al-26Mg extinct radionuclide chronometer to constrain the ages of several recently recovered meteorites that sample previously unknown asteroid lithologies, including the only know felsic meteorite from an asteroid and two other ungrouped basaltic achondrites. These results help broaden our understanding of the timescales involved in igneous differentiation processes in the early SS.