The Type Ia supernova (SN Ia) 2016coj in NGC 4125 (redshift z = 0.00452 ± 0.00006) was discovered by the Lick Observatory Supernova Search 4.9 days after the fitted first-light time (FFLT; 11.1 days before B-band maximum). Our first detection (prediscovery) is merely 0.6 ± 0.5 days after the FFLT, making SN 2016coj one of the earliest known detections of an SN Ia. A spectrum was taken only 3.7 hr after discovery (5.0 days after the FFLT) and classified as a normal SN Ia. We performed high-quality photometry, low- and high-resolution spectroscopy, and spectropolarimetry, finding that SN 2016coj is a spectroscopically normal SN Ia, but the velocity of Si ii λ6355 around peak brightness (∼12,600 kms-1) is a bit higher than that of typical normal SNe. The Si ii λ6355 velocity evolution can be well fit by a broken-power-law function for up to a month after the FFLT. SN 2016coj has a normal peak luminosity (MB ≈ -18.9 ± 0.2 mag), and it reaches a B-band maximum ∼16.0 days after the FFLT. We estimate there to be low host-galaxy extinction based on the absence of Na i D absorption lines in our low- and high-resolution spectra. The spectropolarimetric data exhibit weak polarization in the continuum, but the Si ii line polarization is quite strong (∼0.9% ± 0.1%) at peak brightness.