I start by spending a few minutes with each class, teaching them (or reminding if it is an older class) about the way we do and do not use the brush. I teach them the vocabulary of the brush parts (bristles, ferrule, handle) and remind them that we only paint with the end of the bristles and therefore, I should not see paint on the ferrule, or on the handle. I also talk to them about not "scrubbing" with their brush, so the brush should never look sloppy. (This is a great time to show the video "Young Sloppy Brush" that is found on http://www.teachertube.com/. After this, I pass out the paper and the paints.
For tempera paint, I like to use the "poster paint" containers. You can buy the empty containers from most art supply stores. They hold 6 colors. I use one for the primary and secondaries, and another for the other colors. I like this method of passing out paint because I feel like we waste less paint this way. There are lids that close, and if you need to clean out one color, you can just open that one well, and clean it out. Also, at the end of class, I do not have 20 different palettes to clean laying in the sink. We just put the lids on and go on our way. The containers last a long time, the ones in the pictures are a couple of years old.
I also use "cafeteria" style trays. I place the square brush basins (a must for the classroom in my opinion) in the middle of the tray, and then 2 of the paint containers, on opposite sides of the basin. Next, I place paper towels and brushes (a couple per student.) The great thing about this method of organization is that since the bowls are stable, I am able to stack up 4-5 of the trays on the counter. I can get them ready for the next day. It also makes it much easier to pass out supplies, I can pass everything out in one trip (well, other than their paper), and since the students are not out of their seats, there is less chance of spilling paint or water.
Because I take the time to talk about how to use the brush and paint correctly, and because there is the tray underneath the supplies, there is very little drips and splatters on the table, but if there is, the students have a paper towel in order to clean up the mess right away. I also do not let the students tap their brush on the brush basin to get rid of extra water from their brush. I make them wipe the brush on the side of the bowl.
I use the same set-up for watercolor painting as well.
With kindergartners, I do not let them use watercolors right away. I like to let them use tempera cakes first. I do this because it is the same concept for both types of paint in that they have cakes of color that they have to add water to. The tempera cakes are larger and harder than the watercolor, so the students can learn not to "dig" their brush into the paint, but to add more water. I find that if I have the kindergartners paint with the tempera cakes a couple of times, my watercolors last a lot longer when I finally let the students use them.
Cleanup is very easy, since everything is on the trays. The students place the brushes on the trays, and as I am picking up the trays from their tables, the students are putting their papers on the drying rack. I simply stack the trays up, and clean the supplies when I have a few moments.