From what I remember from medical school (and a quick search of online journals showed no significant newer studies to dispute this), babies can see in color at birth or shortly thereafter (within 1 week). At 1 week, infants are able to discriminate long (e.g. reds, oranges) and medium wavelengths (e.g. yellows, greens) relatively well, reflecting the development of the L cones and M cones in the retina. They have relatively more difficulty discriminating between short wavelengths (e.g. blues, purples). This may be due to absence of the S cones and associated mechanisms in the cerebral cortex. By 2 months, the S cones of infants are functional and by 4 months, infants appear to discriminate color more or less the same way as an adult with normal color vision.
Newborns also have very good contrast discrimination. For example, in the first month, babies can distinguish two shades of gray that differ by only 5% in gray level, and by 2 months, they can see large objects that have less than 0.5% contrast. (Adult contrast sensitivity is 0.2%) On the other hand, a newborn's visual acuity is only about 20/400, at best 20/200 (roughly equivalent to being able to see the big "E" on eye charts).