An optical monitoring survey in nearby dwarf galaxies was carried out with the 2.5m Isaac Newton Telescope. Fifty-five dwarf galaxies and four isolated globular clusters in the Local Group were observed with the Wide Field Camera. The main aims of this survey are to identify the most evolved asymptotic giant branch (AGB) stars and red supergiants at the end-point of their evolution based on their pulsational instability, use their distribution over luminosity to reconstruct the star formation history, quantify the dust production and mass loss from modeling the multiwavelength spectral energy distributions (SEDs), and relate this to luminosity and radius variations. In this first of a series of papers, we present the methodology of the variability survey and describe the photometric catalog of the Andromeda I (And I) dwarf galaxy as an example of the survey, and we discuss the identified long period variable (LPV) stars. We detected 5581 stars and identified 59 LPV candidates within two half-light radii of the center of And I. The amplitudes of these candidates range from 0.2 to 3mag in the i-band. Seventy-five percent of detected sources and 98% of LPV candidates are detected at mid-infrared wavelengths. We show evidence for the presence of dust-producing AGB stars in this galaxy including five extreme AGB (x-AGB) stars, and we model some of their SEDs. A distance modulus of 24.41mag for And I was determined based on the tip of the red giant branch. Also, a half-light radius of 3.2'+/-0.3' was calculated.