A resonantly enhanced photon-regeneration experiment to search for the axion or axionlike particles is described. This experiment is a shining light through walls study, where photons traveling through a strong magnetic field are (in part) converted to axions; the axions can pass through an opaque wall and convert (in part) back to photons in a second region of strong magnetic field. The photon regeneration is enhanced by employing matched Fabry-Perot optical cavities, with one cavity within the axion generation magnet and the second within the photon-regeneration magnet. Compared to simple single-pass photon regeneration, this technique would result in a gain of $(\mathcal{F}/\ensuremath{\pi}{)}^{2}$, where $\mathcal{F}$ is the finesse of each cavity. This gain could feasibly be as high as ${10}^{10}$, corresponding to an improvement in the sensitivity to the axion-photon coupling, ${g}_{a\ensuremath{\gamma}\ensuremath{\gamma}}$, of order $(\mathcal{F}/\ensuremath{\pi}{)}^{1/2}\ensuremath{\sim}300$. This improvement would enable, for the first time, a purely laboratory experiment to probe axion-photon couplings at a level competitive with, or superior to, limits from stellar evolution or solar axion searches. This report gives a detailed discussion of the scheme for actively controlling the two Fabry-Perot cavities and the laser frequencies, and describes the heterodyne signal detection system, with limits ultimately imposed by shot noise.