We present the results from an X-ray and optical survey of a sample of 17 radio jets in active galactic nuclei performed with Chandra and the Hubble Space Telescope. The sample was selected from the radio and is unbiased toward detection at shorter wavelengths, but preferentially it includes beamed sources. We find that X-ray emission is common on kiloparsec scales, with over half (10/17) the radio jets exhibiting at least one X-ray knot on the Chandra images. A similar detection rate is found for the optical emission, although not all X-ray knots have optical counterparts and vice versa. The distributions of the radio-to-X-ray and radio-to-optical spectral indices, α rx and α ro, for the detected jets are similar to the limits for the nondetections, suggesting that all bright radio jets have X-ray counterparts that will be visible in longer observations. Comparing the radio and X-ray morphologies shows that the majority of the X-ray jets have structures that closely map the radio. Analysis of the spectral energy distributions of the jet knots suggests that the knots in which the X-ray and radio morphologies track each other produce X-rays by inverse Compton (IC) scattering of the cosmic microwave background (CMB). The remaining knots produce X-rays by the synchrotron process. Spectral changes are detected along the jets, with the ratio of the X-ray-to-radio and optical-to-radio flux densities decreasing from the inner to the outer regions. This suggests an additional contribution to the X-ray flux in the jet's inner part, either from synchrotron radiation or IC scattering of the stellar light. Alternatively, in a pure IC/CMB scenario, the plasma decelerates as it flows from the inner to the outer regions. Finally, the X-ray spectral indices for the brightest knots are flat (photon index Γ x~ 1.5), indicating that the bulk of the luminosity of the jets is emitted at GeV energies and raising the interesting possibility of future detections with GLAST.