Investigations of HeII Ly{alpha} (304{AA} rest-frame) absorption toward a half-dozen quasars at z~3-4 have demonstrated the great potential of helium studies of the intergalactic medium, but the current critically small sample size of clean sightlines for the HeII Gunn-Peterson test limits confidence in cosmological inferences, and a larger sample is required. Although the unobscured quasar sightlines to high redshift are extremely rare, the Sloan Digital Sky Survey (SDSS) DR6 provides thousands of z>2.8 quasars. We have cross-correlated these SDSS quasars with the Galaxy Evolution Explorer (GALEX) GR2/GR3 to establish a catalog of 200 higher-confidence (~70% secure) cases of quasars at z=2.8-5.1 potentially having surviving far-UV (rest-frame) flux. We also catalog another 112 likely far-UV-bright quasars from GALEX cross-correlation with other (non-SDSS) quasar compilations. Reconnaissance UV prism observations with the Hubble Space Telescope (HST) of 24 of our SDSS/GALEX candidates confirm 12 as detected in the far-UV, with at least nine having flux extending to very near the HeII break; with refinements our success rate is even higher. Our SDSS/GALEX selection approach is thereby confirmed to be an order of magnitude more efficient than previous HeII quasar searches, more than doubles the number of spectroscopically confirmed clean sightlines to high redshift, and provides a resource list of hundreds of high-confidence sightlines for upcoming HeII and other far-UV studies from the HST. Our reconnaissance HST prism spectra suggest some far-UV diversity, confirming the need to obtain a large sample of independent quasar sightlines across a broad redshift range to assess such issues as the epoch(s) of helium reionization, while averaging over individual-object pathology and/or cosmic variance.