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We performed a far-IR imaging survey of the circumstellar dust shells of 144 evolved stars as a mission program of the AKARI infrared astronomical satellite using the Far-Infrared Surveyor (FIS) instrument. With this survey, we deliver far-IR surface brightness distributions of roughly 10′ × 40′ or 10′ × 20′ areas of the sky around the target evolved stars in the four FIS bands at 65, 90, 140, and 160 μm. Our objectives are to characterize the far-IR surface brightness distributions of the cold dust component in the circumstellar dust shells, from which we derive the amount of cold dust grains as low as 20 K and empirically establish the early mass loss history. In this first installment of the series, we introduce the project and its aims, describe the observations, data reduction, and surface brightness correction process, and present the entire data set along with the results of integrated photometry measurements (i.e. the central source and circumstellar dust shell together). We find that: (1) far-IR emission is detected from all but one object at the spatial resolution about 30′–50′ in the corresponding bands; (2) roughly 60%–70% of the target sources show some extension; (3) previously unresolved nearby objects in the far IR are now resolved around 28 target sources; (4) the results of photometry measurements are reasonable with respect to the entries in the AKARI/FIS Bright Source Catalogue, despite the fact that the targets were assumed to be point sources when catalogue flux densities were computed; and (5) an IR two-color diagram would place the target sources in a roughly linear distribution that may correlate with the age of the circumstellar dust shell and can potentially be used to identify which targets are more extended than others. [ABSTRACT FROM AUTHOR]