Pulsar timing has been an irreplaceable technique in studies of gravitational radiation for decades and is now poised to allow direct detection and investigation of nanohertz gravitational waves (GWs). The most promising sources of GWs detectable by pulsar timing arrays (PTAs) are supermassive black hole binaries (SMBHBs), making PTAs indispensable in studies of galaxy mergers and evolution. This dissertation aims to describe several new techniques developed to facilitate the detection of a variety of GW signals by PTAs. Recognizing modern advances in astrometric measurements of millisecond pulsars (MSPs) with very long baseline interferometry (VLBI), we have systematically investigated the advantageous role such astrometric measurements can play in PTA searches for GWs, especially in searches for a stochastic background of GWs. We describe a detailed program by which VLBI astrometry of MSPs can be incorporated into PTA efforts to detect GWs, addressing issues related to mismatched coordinate systems between the VLBI and pulsar timing communities and the time necessary to make VLBI measurements. Memory, a permanent change in the spacetime surrounding a powerful GW source, is a potentially prevalent type of GW detectable by PTAs. We develop a variety of methods to detect such signals or to place upper limits on their rates at various amplitudes. We apply our techniques in a search for GW bursts with memory (BWMs) in the first five years of data from the North American Nanohertz Observatory for Gravitational Waves (NANOGrav). Finally, we describe a method by which GWs from single point-like sources with any time-domain behavior can be isolated and studied with PTA data sets. These techniques rely solely on the assumption that GWs are quadrupolar and occur in two polarization modes. We demonstrate the flexibility of these techniques in several explanatory examples.