The NANOGrav Collaboration found strong evidence for a common-spectrum stochastic process in our 12.5-yr pulsar timing array (PTA) dataset, with median characteristic strain amplitude at periods of a year of Ayr = 1.92+0.75-0.55 × 10-15. However, evidence for the quadrupolar Hellings & Downs interpulsar correlations, which are characteristic of gravitational wave (GW) signals, was not significant. We emulate and extend the NANOGrav dataset, injecting a wide range of stochastic gravitational wave background (SGWB) signals that encompass a variety of amplitudes and spectral shapes. We then apply our standard detection pipeline and explore three key astrophysics milestones: (I) robust detection of the SGWB; (II) determination of the source of the SGWB; and, (III) measurement of the shape of the SGWB spectrum. Given the amplitude measured in the 12.5 yr analysis and assuming this signal is a SGWB, we expect to accumulate robust evidence of an interpulsar-correlated SGWB signal with 15–17 yrs of data. At the initial detection, we expect a fractional uncertainty (bounding the 95% credible region) of 40% on the power-law strain spectrum slope, which is sufficient to distinguish a SGWB of supermassive black-hole binary origin from some models predicting primordial or cosmic-string origins. Similarly, the measured SGWB amplitude will have an uncertainty of 44% upon initial detection, allowing us to arbitrate some population models of supermassive black-hole binaries. We also validate the efficacy of assuming a power-law SGWB strain spectrum for recovering realistic signals, even when significant attenuation at low frequencies is present. In general, models that produce low-frequency spectral turnovers are distinguishable with 20~yrs of data. Even though our study is based on the NANOGrav data and we phrase our main results with respect to it, we also show relations that allow for a generalization to other PTA datasets. Most notably, by combining individual PTA’s data into the International Pulsar Timing Array, all of these milestones can be reached significantly earlier.