Squeeze out as much paint from the bristles as possible before adding any sort of cleanser, wiping the brush on rags or newspaper. This makes it easier to wash the bristles clean in future steps.
Choose the right cleanser for your medium. Oil paints require the use of paint thinner or mineral spirits to remove the majority of the paint before switching to a liquid soap to finish the job. Acrylic paints may be water-based, but the acrylic binder can be a little tough for just water to handle, so a commercial brush cleaner comes in handy. Watercolors are one of the few mediums that is truly easy to clean up with just clean water.
Continue to clean and rinse the brush until the water runs absolutely clear and no more pigment shows up in your soap or cleanser.
Dry your brushes with the bristles pointing down so the water drains out instead of into the handle. Even a slight incline created by resting the handle on a towel or book will be sufficient. Trapped water can warp the wood of the handle or loosen the bristles from their barrel, destroying the brush for future use.
Reshape the tips of your brushes before allowing them to dry. Some brushes are stiff and keep their shape with no problem, but others are more delicate and will benefit from a moment's care in reshaping a point or bevel.
Mount a metal spring or coil to the edge of a cabinet or ledge with a staple gun (or on a movable shelf if you do not have a dedicated sink in your studio). This lets you dry the brushes, bristles down, without worrying about them sitting on the bristles and becoming deformed or warped. Just slip the handle between the coils of the metal to hold it until dry.
Place the brushes, handles down, in a jar, vase or canister beside your easel only if you are a prolific painter and will be using them a lot. Make sure they are completely dry before putting them away if you store them in this manner.
Keep the brushes lying flat in a draw, roll-up caddy or case for more occasional use. This keeps dust away from the bristles and protects them as long as you don't overcrowd the drawer or case.
Reuse the plastic tubes that come with some finer brushes to protect the bristles between use. Just slip it back over the brush and push down gently, but firmly enough to make it stay.
Mount pipettes with the end of the bulb removed or open test tubes horizontally, using packing tape, to an area within easy reach of your workspace, and place the brushes, one or two in each if using test tubes (cigar tubes work well too), inside to make them available when needed, but out of the way when not. This is especially good for small brushes and small workspaces.