We present photometry and spectroscopy of four hydrogen-poor luminous supernovae discovered during the 2-month long science commissioning and early operations of the Zwicky Transient Facility (ZTF) survey. Three of these objects, SN 2018bym (ZTF18aapgrxo), SN 2018avk (ZTF18aaisyyp), and SN 2018bgv (ZTF18aavrmcg), resemble typical SLSN-I spectroscopically, while SN 2018don (ZTF18aajqcue) may be an object similar to SN 2007bi experiencing considerable host galaxy reddening, or an intrinsically long-lived, luminous, and red SN Ic. We analyze the light curves, spectra, and host galaxy properties of these four objects and put them in context of the population of SLSN-I. SN 2018bgv stands out as the fastest-rising SLSN-I observed to date, with a rest-frame g-band rise time of just 10 days from explosion to peak—if it is powered by magnetar spin-down, the implied ejecta mass is only 1 M _⊙. SN 2018don also displays unusual properties—in addition to its red colors and comparatively massive host galaxy, the light curve undergoes some of the strongest light-curve undulations postpeak seen in an SLSN-I, which we speculate may be due to interaction with circumstellar material. We discuss the promises and challenges of finding SLSNe in large-scale surveys like ZTF given the observed diversity in the population.