Atomic hydrogen traces the raw material from which molecular clouds and stars form. With 565 galaxies from the ALFALFA Hα survey, a statistically complete subset of the ALFALFA survey, we examine the processes that affect galaxies' abilities to access and consume their Hi gas. On galaxy-wide scales, Hi gas fractions correlate only weakly with instantaneous specific star formation rates (sSFRs) but tightly with galaxy color. We show that a connection between dust and Hi content, arising from the fundamental mass-metallicity-Hi relation, leads to this tight color correlation. We find that disk galaxies follow a relation between stellar surface density and Hi depletion time, consistent with a scenario in which higher mid-plane pressure leads to more efficient molecular cloud formation from Hi. In contrast, spheroids show no such trend. Starbursts, identified by Hα equivalent width, do not show enhanced Hi gas fractions relative to similar mass non-starburst galaxies. The starbursts' shorter Hi depletion times indicate more efficient consumption of Hi, and galaxy interactions drive this enhanced star formation efficiency in several starbursts. Interestingly, the most disturbed starbursts show greater enhancements in Hi gas fraction, which may indicate an excess of Hi at early merger stages. At low galaxy stellar masses, the triggering mechanism for starbursts is less clear; the high scatter in efficiency and sSFR among low-mass galaxies may result from periodic bursts. We find no evidence for depleted Hi reservoirs in starbursts, which suggests that galaxies may maintain sufficient Hi to fuel multiple starburst episodes.