We present optical photometry of XTE J2123-058 taken in 1999 and 2000, during its quiescent state. The dominant feature of our R-band light curve is the ellipsoidal modulation of the secondary star; however, in order to fit this satisfactorily, we require additional components that comprise an X-ray-heated Roche lobe-filling secondary star and an accretion disk bulge, i.e., where the gas stream impacts the accretion disk. The observed dip near phase 0.8 is interpreted as the eclipse of inner parts of the accretion disk by the bulge. This scenario is highly plausible given the high binary inclination. Our fits allow us to constrain the size of the quiescent accretion disk to lie in the range (0.26-0.58)RL1 (68% confidence). Using the distance of 9.6 kpc and the X-ray flux inferred from the heated hemisphere of the companion, we obtain an unabsorbed X-ray luminosity of 1.2×1033 ergs s-1 for XTE J2123-058 in quiescence. From the observed quiescent optical/IR colors, we find that the power-law index (-1.4) for the spectral distribution of the accretion disk compares well with other quiescent X-ray transients. We also reanalyze the optical light curves of the soft X-ray transient XTE J2123-058 taken during its outburst and decay in 1998. We use a robust method to fit the data using a refined X-ray binary model. The model computes the light arising from a Roche lobe-filling star and flared accretion disk irradiated by X-rays and calculates the effects of shadowing and mutual star/disk eclipses. We obtain relatively accurate values for the binary inclination and mass ratio, which when combined with spectroscopic results obtained in Paper II, gives a neutron star mass in the range 1.04-1.56 Msolar (68% confidence).