Using data collected with the CLEO detector operating at the CESR e+e- collider at sqrt[s]=3.97-4.26 GeV, we investigate 15 charmonium decay modes of the psi(4040), psi(4160), and Y(4260) resonances. We confirm, at 11 sigma significance, the BABAR Y(4260)-->pi+pi- J/psi discovery, make the first observation of Y(4260)--> pi(0)pi(0) J/psi (5.1 sigma), and find the first evidence for Y(4260)-->K+K- J/psi(3.7 sigma). We measure e+e- cross sections at sqrt[s]=4.26 GeV as sigma(pi+pi- J/psi)=58(+12)(-10)+/-4 pb, sigma(pi(0)pi(0) J/psi)=23(+12)(-8)+/-1 pb, and sigma(K+K- J/psi)=9(+9)(-5)+/-1 pb, in which the uncertainties are statistical and systematic, respectively. Upper limits are placed on other decay rates from all three resonances.