Richard Feynman spent most of his career at Caltech, formally the California Institute of Technology. He joined the institute in 1952 and remained there for the rest of his academic career.
Before Caltech, Feynman taught at Cornell University from 1945 to 1950. He then spent a short period at the Institute for Advanced Study in Princeton before moving to California. At Caltech, he held the Richard C. Tolman Professorship of Physics and became one of the institution's most influential teachers and researchers.
His Caltech work included research on quantum electrodynamics, superfluidity, particle physics, and the path-integral formulation of quantum mechanics. He also helped create The Feynman Lectures on Physics, based on his undergraduate teaching in the early 1960s.
MIT and Princeton were important parts of Feynman's education: he earned his undergraduate degree at MIT and his doctorate at Princeton. Cornell was his major pre-Caltech academic appointment, but Caltech is the canonical answer for where he spent most of his career.