We have searched for H I, CO, and HCO+ in absorption along 16 lines of sight toward high-velocity clouds (HVCs). Of these, seven lines of sight are toward true HVCs, two are toward the Magellanic Stream, six are toward the outer arm complex, and one is toward a probable Galactic cloud. We detected absorption along two lines of sight toward the outer arm complex, at T-S = 38 and 147 K, respectively. The detections are consistent with normal disk H I absorption. The nondetections suggest that HVCs are weak H I absorbers and that they contain primarily warm H I, with a surface filling fraction of cold (similar to 50 K) gas of less than 7%. In one case, we obtained a 3 sigma lower limit to the spin temperature of 180-320 K. Upper limits for CO and HCO+ emission along seven lines of sight are the most sensitive probes of molecular emission yet toward HVCs. The best upper limits imply N(CO) < 5.5 x 10(14) and N(H-2) < 5.5 x 10(18) cm(-2), assuming solar abundances and in one case a molecular gas fraction less than 17% at the 3 sigma level. Averaging the CO nondetections to test for broader, low-level absorption implies N(CO) < 4.8 x 10(14) cm(-2).