We present the detections of shocked molecular hydrogen (H2) gas in near- and mid-infrared and broad CO in millimeter from the mixed-morphology supernova remnant (SNR) HB~3 (G132.7+1.3) using Palomar WIRC, the Spitzer GLIMPSE360 and WISE surveys, and HHSMT. Our near-infrared narrow-band filter H2 2.12 micron images of HB~3 show that both Spitzer IRAC and WISE 4.6 micron emission originates from shocked H2 gas. The morphology of H2 exhibits thin filamentary structures and a large scale of interaction sites between the HB~3 and nearby molecular clouds. Half of HB~3, the southern and eastern shell of the SNR, emits H2 in a shape of a butterfly or W, indicating the interaction sites between the SNR and dense molecular clouds. Interestingly, the H2 emitting region in the southeast is also co-spatial to the interacting area between HB~3 and the H~II regions of the W3 complex, where we identified star-forming activity. We further explore the interaction between HB~3 and dense molecular clouds with detections of broad CO(3-2) and CO(2-1) molecular lines from the southern and southeastern shells along the H2 emitting region. The widths of the broad lines are 8-20 km/s; the detection of such broad lines is unambiguous, dynamic evidence of the interactions between the SNR and clouds. The CO broad lines are from two branches of the bright, southern H2 shell. We apply the Paris-Durham shock model to the CO line profiles, which infer the shock velocities of 20 - 40 km/s, relatively low densities of 10^{3-4} cm^{-3} and strong (>200 micro Gauss) magnetic fields.
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