What do colored noises look like in the time domain?
White noise implies no correlation between samples of the noise, even consecutive samples.  Colored noise, therefore, implies that there is correlation of some sort between the noise samples, which in turn implies that we can take advantage of that correlation to get rid of some of the noise.

Beyond that, there is not a lot that we can say about what it looks like in the time domain other than to say that colors that favor low frequencies, like pink noise, will tend to change more slowly, while high frequency colors, like "blue" noise, will tend to change more quickly.