We measure the ratio of branching fractions, R0 = B(D0 -> pi l nu)/B(D0 -> K l nu), where l is an electron or muon. We also measure the normalized rates for D0 -> pi l nu and D0 -> K l nu as a function of the momentum transfer squared (q^2). The data were collected with the CLEO-III detector while the Cornell Electron Storage Ring (CESR) was running at or just under the Y(4S) resonance. We find R0 = 0.082 +- 0.006 +- 0.005, where the first uncertainty is statistical and the second is systematic. We fit the normalized q^2 distributions with two form factor parameterizations, the simple pole and modified pole functions. For the simple pole fits, we find effective masses, M_poleDK = 1.89 +0.05 -0.05 +0.04 -0.03 GeV and M_poleDpi = 1.86 +0.10 -0.06 +0.07 -0.03 GeV, where the first uncertainty is statistical and the second is systematic. For the modified pole fits, we find alpha_DK = 0.36 +0.11 -0.10 +0.03 -0.07 and alpha_Dpi = 0.37 +0.20 -0.31 +0.16 -0.14, where (1/sqrt(alpha))M_D(s) is the effective mass of the second pole and again the first uncertainty is statistical and the second is systematic. Using the measured value of R0, and the modified-pole-optimized form factor shapes, we find the product of the ratio of form factor normalizations at q^2 = 0 and the ratio of CKM matrix elements, |f^pi_+(0)/f^K_+(0)|^2|Vcd/Vcs|^2 = 0.038+0.006 -0.007 +0.005 -0.003 (the first uncertainty is statistical and the second is systematic). Experimental measurements of these heavy-to-light form factors are crucial to constraining the theoretical techniques used to calculate them. Improved understanding of semileptonic form factors can aid precision measurements of the CKM matrix elements, in particular exclusive semileptonic measurements of |Vub|.