We have observed the 5.54 s anomalous X-ray pulsar XTE J1810-197 at radio, millimeter, and infrared (IR) wavelengths, with the aim of learning about its broadband spectrum. At the IRAM 30 m telescope, we have detected the magnetar at ν=88 and 144 GHz, the highest radio-frequency emission ever seen from a pulsar. At 88 GHz we detected numerous individual pulses, with typical widths ~2 ms and peak flux densities up to 45 Jy. Together with nearly contemporaneous observations with the Parkes, Nançay, and Green Bank telescopes, we find that in late 2006 July the spectral index of the pulsar was -0.5<~α<~0 (with flux density Sν~να) over the range 1.4-144 GHz. Nine dual-frequency Very Large Array and Australia Telescope Compact Array observations in 2006 May-September are consistent with this finding, while showing variability of α with time. We infer from the IRAM observations that XTE J1810-197 remains highly linearly polarized at millimeter wavelengths. Also, toward this pulsar, the transition frequency between strong and weak scattering in the interstellar medium may be near 50 GHz. At Gemini, we detected the pulsar at 2.2 μm in 2006 September, at the faintest level yet observed, Ks=21.89+/-0.15. We have also analyzed four archival IR Very Large Telescope observations (two unpublished), finding that the brightness fluctuated within a factor of 2-3 over a span of 3 years, unlike the monotonic decay of the X-ray flux. Thus, there is no correlation between IR and X-ray flux, and it remains uncertain whether there is any correlation between IR and radio flux.