We report the initial results of the Brown Dwarf Spectroscopic Survey Gravity Project to study gravity sensitive features as indicators of youth in brown dwarfs. Low-resolution (R similar to 2000) J-band and optical (R similar to 1000) observations using NIRSPEC and LRIS at the W. M. Keck Observatory reveal transitions of TiO, VO, K I, Na I, Cs I, Rb I, CaH, and FeH. By comparing these features in late-type giants and in old field dwarfs, we show that they are sensitive to the gravity (g = GM/R-2) of the object. Using low-gravity spectral signatures as age indicators, we observed and analyzed J-band and optical spectra of two young brown dwarfs, G196-3B (20-300 Myr) and KPNO Tau 4 (1-2 Myr) and two possible low-mass brown dwarfs in the sigma Orionis cluster (3-7 Myr). We report the identification of the phi bands of TiO near 1.24 mum and the A-X band of VO near 1.18 mum together with extremely weak J-band lines of K I in KPNO Tau 4. This is the first detection of TiO and VO in the J band in a substellar-mass object. The optical spectrum of KPNO Tau 4 exhibits weak K I and Na I lines, weak absorption by CaH, and strong VO bands, also signatures of a lower gravity atmosphere. G196-3B shows absorption features in both wavelength regions, like those of KPNO Tau 4, suggesting that its age and mass are at the lower end of published estimates. Whereas sigma Ori 51 appears to be consistent with a young substellar object, sigma Ori 47 shows signatures of high gravity most closely resembling an old L1.5/L0 and cannot be a member of the sigma Orionis cluster.