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 ~ 2000)J-band and optical (R ~ 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/R²) of the object. Using low-gravity spectral signatures as age indicators, we observed and analyzedJ-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 σ Orionis cluster (3-7 Myr). We report the identification of the ϕ bands of TiO near 1.24 μm and the A-X band of VO near 1.18 μm together with extremely weakJ-band lines of K I in KPNO Tau 4. This is the first detection of TiO and VO in theJ 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 σ Ori 51 appears to be consistent with a young substellar object, σ Ori 47 shows signatures of high gravity most closely resembling an old L1.5/L0 and cannot be a member of the σ Orionis cluster.