We explore the multidimensional, multiwavelength selection of quasars from mid-infrared (MIR) plus optical data, specifically from Spitzer-Infrared Array Camera (IRAC) and the Sloan Digital Sky Survey (SDSS). Traditionally, quasar selection relies on cuts in two-dimensional color space despite the fact that most modern surveys (optical and IR) are done in more than three bandpasses. In this paper, we apply modern statistical techniques to combined Spitzer MIR and SDSS optical data, allowing up to eight-dimensional (8-D) color selection of quasars. Using a Bayesian selection method, we catalog 5546 quasar candidates to an 8.0 μm depth of 56 μJy over an area of ~24 deg2. Roughly 70% of these candidates are not identified by applying the same Bayesian algorithm to 4-color SDSS optical data alone. The 8-D optical+MIR selection on this data set recovers 97.7% of known type 1 quasars in this area and greatly improves the effectiveness of identifying 3.5 < z < 5 quasars which are challenging to identify (without considerable contamination) using MIR data alone. We demonstrate that, even using only the two shortest wavelength IRAC bandpasses (3.6 and 4.5 μm), it is possible to use our Bayesian techniques to select quasars with 97% completeness and as little as 10% contamination (as compared to ~60% contamination using color cuts alone). We compute photometric redshifts for our sample; comparison with known objects suggests a photometric redshift accuracy of 93.6% (Δz ± 0.3), remaining roughly constant when the two reddest MIR bands are excluded. Despite the fact that our methods are designed to find type 1 (unobscured) quasars, as many as 1200 of the objects are type 2 (obscured) quasar candidates. Coupling deep optical imaging data, with deep MIR data, could enable selection of quasars in significant numbers past the peak of the quasar luminosity function (QLF) to at least z ~ 4. Such a sample would constrain the shape of the QLF both above and below the break luminosity (L* Q ) and enable quasar clustering studies over the largest range of redshift and luminosity to date, yielding significant gains in our understanding of the physics of quasars and their contribution to galaxy evolution.