Cyclic codes are among the most important families of codes in coding theory for both theoretical and practical reasons. Despite their prominence and intensive research on cyclic codes for over a half century, there are still open problems related to cyclic codes. In this work, we use recent results on the equivalence of cyclic codes to create a more efficient algorithm to partition cyclic codes by equivalence based on cyclotomic cosets. This algorithm is then implemented to carry out computer searches for both cyclic codes and quasi-cyclic (QC) codes with good parameters. We also generalize these results to repeated-root cases. We have found several new linear codes that are cyclic or QC as an application of the new approach, as well as more desirable constructions for linear codes with best known parameters. With the additional new codes obtained through standard constructions, we have found a total of 14 new linear codes.