The interstellar oxygen isotopic ratio of ^18^O/^17^O can reflect the relative amount of the secular enrichment by ejecta from high-mass versus intermediate-mass stars. Previous observations found a Galactic gradient of ^18^O/^17^O, i.e., low ratios in the Galactic center and large values in the Galactic disk, which supports the inside-out formation scenario of our Galaxy. However, there are not many observed objects and, in particular, there are not many at large galactocentric distances. For this reason, we started a systematic study on Galactic interstellar ^18^O/^17^O, through observations of C^18^O and C^17^O multi-transition lines toward a large sample of 286 sources (at least one order of magnitude larger than previous ones), from the Galactic center region to the far outer Galaxy (~22kpc). In this article, we present our observations of J=1-0 lines of C^18^O and C^17^O, with the 12m antenna of the Arizona Radio Observatory (ARO 12m) and the Institut de Radio Astronomie Millimetrique (IRAM) 30m telescopes. Among our IRAM 30m sample of 50 targets, we detected successfully both C^18^O and C^17^O 1-0 lines for 34 sources. Similarly, our sample of 260 targets for ARO 12m observations resulted in the detection of both lines for 166 sources. The C^18^O optical depth effect on our ratio results, evaluated by fitting results of C^17^O spectra with hyperfine components (assuming {tau}_C18O_=4{tau}_C17O_) and our radiative transfer and excitation model nonlocal thermodynamic equilibrium (non-LTE) model calculation for the strongest source, was found to be insignificant. Beam dilution does not seem to be a problem either, which was supported by the fact that there is no systematic variation between the isotopic ratio and the heliocentric distance, and ratios are consistently measured from two telescopes for most of those detected sources. With this study we obtained ^18^O/^17^O isotopic ratios for a large sample of molecular clouds with different galactocentric distances. Our results, though there are still very few detections made for sources in the outer Galaxy, confirm the apparent ^18^O/^17^O gradient of ^18^O/^17^O=(0.10+/-0.03)R_GC_+(2.95+/-0.30), with a Pearson's rank correlation coefficient of R=0.69. This is supported by the newest Galactic chemical evolution model including the impact of massive stellar rotators and novae.