The Neil Gehrels Swift Observatory carried out prompt searches for gravitational-wave (GW) events detected by the LIGO/Virgo Collaboration (LVC) during the second observing run ("O2"). Swift performed extensive tiling of eight LVC triggers, two of which had very low false-alarm rates (GW170814 and the epochal GW170817), indicating a high confidence of being astrophysical in origin; the latter was the first GW event to have an electromagnetic counterpart detected. In this paper we describe the follow-up performed during O2 and the results of our searches. No GW electromagnetic counterparts were detected; this result is expected, as GW170817 remained the only astrophysical event containing at least one neutron star after LVC's later retraction of some events. A number of X-ray sources were detected, with the majority of identified sources being active galactic nuclei. We discuss the detection rate of transient X-ray sources and their implications in the O2 tiling searches. Finally, we describe the lessons learned during O2 and how these are being used to improve the Swift follow-up of GW events. In particular, we simulate a population of gamma-ray burst afterglows to evaluate our source ranking system's ability to differentiate them from unrelated and uncataloged X-ray sources. We find that ~60%-70% of afterglows whose jets are oriented toward Earth will be given high rank (i.e., "interesting" designation) by the completion of our second follow-up phase (assuming that their location in the sky was observed), but that this fraction can be increased to nearly 100% by performing a third follow-up observation of sources exhibiting fading behavior.