He hoped that he would learn to quickly run so that he could win the race, and be the best runner

Explanation

In English, adverbs are usually placed before the main verb but after auxiliary verbs (such as be, have, may, & must). The only exception is when the main verb is "to be", in which case the adverb goes after the main verb. So, here the adverb (quickly) will be used after because the main verb is "to run". Therefore, it should be "to run quickly" instead of "to quickly run. The error is in the placement of the adverb "quickly." The traditional rule is not to split an infinitive by placing an adverb between "to" and the base verb. The correct placement is "to run quickly".