Water alone does not help clean off oil paint, and you would not want to wet your brushes in between colors either. Water works well with acrylics, but water and oil don't like each other.
There are two main ways to clean a brush before loading it with another color.
Before I get into that, let me mention that one way to minimize brush cleaning is to keep different brushes for different colors. For example, you may keep one brush for all very light colors, one for the dark and grayish colors, one for the reds, and one for the greens.
1. When you switch to a different color, the first thing to do is to squeeze out all the paint from the brush using a paper towel or a rag. Squeeze the bristles in the rag, applying pressure with your fingers. You may also rub the brush on it to get most of the paint out.
There will be a little of the old color still in the brush, but if you use it for a similar color, the problem is minimal. Just be aware how you apply the paint. If you rub the brush on the canvas, the old color will come out as well.
2. The second way to clean your brushes is to use thinner. I use odorless turpentine to thin my paint and, on occasion, when I need a good brush cleaning during a painting session, I dip my brush in the thinner and then clean it well using a rag. Before doing this, you want to remove all the paint you can with a dry rag or paper towel. Use the mineral spirit as the last step of thorough cleaning. Some people try to avoid the thinner as it's full of pollutants, and instead do this with safflower oil. That works too; the procedure is the same as with the thinner.
To clean my brushes at the end of a painting session I use water and soap. I get the bristles all soapy and then rub them on the palm of my (gloved) hand, rinse a little, and repeat until I see no more paint coming out from the brush. Then I rinse well, squeeze the excess water out, and I lay that brushes flat to dry until the next day.