We outline our GRB afterglow observation program using the 1-m telescope at Mt. Lemmon Optical Astronomy Observatory (LOAO), and report the first observations of the GRB afterglows. During the 2007B semester, we performed follow-up imaging obsrevations of 6 GRBs, and succeeded in detecting four GRB afterglows (GRB 071010B, GRB 071018, GRB 071020, and GRB 071025) while placing useful upper limits on the light curves of the other GRBs. Among the observed events, we find that three events are special and interesting. GRB 071010B has a light curve which has an unusually long jet break time of 11.8 days. For GRB 071025, its red R-I(~2) color suggests that it is likely to be at z~5. GRB 071020 has a light curve which shows a clear brightening at 0.3-1 days after the burst, where our LOAO data play a crucial role by providing an unambiguous evidence for the brightening. These are the first successful detections of GRB afterglows by a facility owned and operated by a Korean institution, demonstrating the usefulness of the 1-m telescope for transient phenomena such as GRBs up to very high redshift.