We describe new measurements of the inclusive and exclusive branching fractions for psi(2S) transitions to J/psi using e(+)e(-) collision data collected with the CLEO detector operating at CESR. All branching fractions and ratios of branching fractions reported here represent either the most precise measurements to date or the first direct measurements. Indirectly and in combination with other CLEO measurements, we determine B(chi(cJ) --> gamma(J/psi)) and B[psi(2S) --> light hadrons].