Doing messy art with kids prepares them for independence at an early age. Kids get used to keeping the creative process in a designated spot, be it the kitchen or a specific area. When they are very young, they watch you clean up. Describe what you are doing and try to make clean-up fun too. I used to sit the kids on the kitchen counter to wipe their painted faces. The mess is part of the process. If you gripe too much, they may start avoiding messy art.
As they become toddlers they can clean with you side-by-side, washing brushes together in a sink of soapy water. They can put paint containers back in a big box. Kids gradually learn to clean up independently if they take an active role. It will simply become part of creating art.
At 6 and 8, my daughters have the process down. They know that clay and wet projects need to happen in the kitchen at a table that we bring in from an adjacent room. They know how to clean brushes, to spray the table with soap & water and wipe with old towells.
Most other projects are done at the breakfast room table. The breakfast room has a window ledge full of clear boxes which contain art supplies that are used daily. These include plain and patterned paper, glue, tape, stapler, scissors, sets of markers (ultra-thin Sharpies, Papermate markers, colored pencils), rulers, plus boxes for each kid to stash flat work-in-process, coloring books, notebooks, journals, anything they use to create art or write stories. For projects with small components, such as beads, the kids grab a metal loaf pan or cake pan from the pantry, or use plastic storage containers, to keep the mess from falling all over the floor. Our dog likes to eat beads, very crunchy. When they start a project, they get out all of the materials and set up the work space, similar to the way they do tasks at school.
On 43things, I searched for "organize art supplies" and there were 8,686 people sharing the goal. Having art supplies accessible is key to actually using them! To keep all of the project-related art and craft supplies, My husband installed two sets of shelves in our pantry and we use one side for food-related stuff and the other for art supplies.
I store art supplies by category in clear boxes labeled with Sharpies. Watercolor pans, tube watercolors and brushes are stored together. Acrylics, brushes and sponges are together. Beads are with bead string and tools. Clay is kept with shaping tools. String and ribbon are together. Rubber stamps are stored with stamp pads. Supplies for hand sewing, such as cut fabric, scissors, safety pins and measuring tape are kept together. The idea is to be able to pull out a box and have just about everything that you need to do your project. There are some things that I keep together which are used on many different projects, such as glue, glitter, things that can be attached to things, felt, stickers, scissors and tape.