Observing nearby galaxies with submillimeter telescopes on the ground has two major challenges. First, the brightness is significantly reduced at long submillimeter wavelengths compared to the brightness at the peak of the dust emission. Second, it is necessary to use a high-pass spatial filter to remove atmospheric noise on large angular scales, which has the unwelcome side effect of also removing the galaxy's large-scale structure. We have developed a technique for producing high-resolution submillimeter images of galaxies of large angular size by using the telescope on the ground to determine the small-scale structure (the large Fourier components) and a space telescope (Herschel or Planck) to determine the large-scale structure (the small Fourier components). Using this technique, we are carrying out the HARP and SCUBA-2 High Resolution Terahertz Andromeda Galaxy Survey (HASHTAG), an international Large Program on the James Clerk Maxwell Telescope, with one aim being to produce the first high-fidelity high-resolution submillimeter images of Andromeda. In this paper, we describe the survey, the method we have developed for combining the space-based and ground-based data, and we present the first HASHTAG images of Andromeda at 450 and 850 mu m. We also have created a method to predict the CO(J = 3-2) line flux across M31, which contaminates the 850 mu m band. We find that while normally the contamination is below our sensitivity limit, it can be significant (up to 28%) in a few of the brightest regions of the 10 kpc ring. We therefore also provide images with the predicted line emission removed.