We present ultraviolet (UV) and optical photometry of 26 Type Ia supernovae (SNe Ia) observed from 2005 March to 2008 March with the NASA Swift Ultraviolet and Optical Telescope (UVOT). The dataset consists of 2133 individual observations, making it by far the most complete study of the UV emission from SNe Ia to date. Grouping the SNe into three subclasses as derived from optical observations, we investigate the evolution of the colors of these SNe, finding a high degree of homogeneity within the normal subclass, but dramatic differences between that group and the subluminous and SN 2002cx-like groups. For the normal events, the redder UV filters on UVOT (u, uvw1) show more homogeneity than do the bluer UV filters (uvm2, uvw2). Searching for purely UV characteristics to determine existing optically based groupings, we find the peak width to be a poor discriminant, but we do see a variation in the time delay between peak emission and the late, flat phase of the light curves. The UV light curves peak a few days before the B band for most subclasses (as was previously reported by Jha et al.), although the SN 2002cx-like objects peak at a very early epoch in the UV. That group also features the bluest emission observed among SNe Ia. As the observational campaign is ongoing, we discuss the critical times to observe, as determined by this study, in order to maximize the scientific output of future observations.