  We present high-precision timing observations spanning up to nine years for
37 millisecond pulsars monitored with the Green Bank and Arecibo radio
telescopes as part of the North American Nanohertz Observatory for
Gravitational Waves (NANOGrav) project. We describe the observational and
instrumental setups used to collect the data, and methodology applied for
calculating pulse times of arrival; these include novel methods for measuring
instrumental offsets and characterizing low signal-to-noise ratio timing
results. The time of arrival data are fit to a physical timing model for each
source, including terms that characterize time-variable dispersion measure and
frequency-dependent pulse shape evolution. In conjunction with the timing model
fit, we have performed a Bayesian analysis of a parameterized timing noise
model for each source, and detect evidence for excess low-frequency, or "red,"
timing noise in 10 of the pulsars. For 5 of these cases this is likely due to
interstellar medium propagation effects rather than intrisic spin variations.
Subsequent papers in this series will present further analysis of this data set
aimed at detecting or limiting the presence of nanohertz-frequency
gravitational wave signals.
