Regenerating codes (RCs) can significantly reduce the repair-bandwidth of distributed storage networks. Initially, the analysis of RCs was based on the assumption that during the repair process, the newcomer does not distinguish (among all surviving nodes) which nodes to access, i.e., the newcomer is oblivious to the set of helpers being used. Such a scheme is termed the blind helper selection (BHS) scheme. Nonetheless, it is intuitive in practice that the newcomer should choose to access only those "good" helpers. In this two-part paper, a new characterization of the effect of choosing the helper nodes in terms of the storage-bandwidth tradeoff is given. Specifically, the answer to the following fundamental question is provided: Under what condition does proactively choosing the helper nodes improve the storage-bandwidth tradeoff? Through a graph-based analysis, this Part I paper answers this question by providing a necessary and sufficient condition under which optimally choosing good helpers strictly improves the storage-bandwidth tradeoff. A low-complexity helper selection solution, termed the family helper selection (FHS) scheme, is proposed and the corresponding storage/repair-bandwidth curve is characterized. This Part I paper also proves that under some design parameters, the FHS scheme is indeed optimal among all helper selection schemes. In the Part II paper, an explicit construction of an exact-repair code is proposed that achieves the minimum-bandwidth-regenerating (MBR) point of the FHS scheme. The new exact-repair code can be viewed as a generalization of the existing fractional repetition code.