Glaciergus has a good writeup. I'll just include mine from a manual that I put together a few years ago for my wife (she has yet to dump the tanks so not sure why I bothered)...
Here is the cliff notes version ...
1. Warning - Do not open the valves until you are ready. Use disposable gloves.
2. Remove the holding tank outlet cap and connect the three-inch sewer hose to the outlet of the holding tank.
3. Extend the hose to the hole in the ground.
4. Insert the sewer hose into the dump station's hole a few inches. Use the hole's cover, a rock, or something heavy enough to hold the sewer hose in place so it doesn't pop out of the hole.
5. Open the black tank valve first. You'll hear the "stuff" flow. This will usually take a few minutes depending on how long you have been camping.
6. Close the black tank valve.
7. Open the gray tank valve. Drain until its stopped.
8. Close the gray tank valve.
9. Check both black and gray water tank valves and make sure they are closed.
10. Disconnect the sewer hose from the outlet and replace the outlet cover.
11. Rinse the hose with the Not-Potable Water Hose and store away in the bumper.
12. Its a must to add 3-5 gallons of water to the black tank by flushing the toilet and then add the water softener (we use the Geo Method).
Granny - That's the basics. There are a number of videos on youtube that might be worth watching, as well.
Some will flush the system if they are equipped with a flush. The clear plastic elbow you see in that picture is nice so that you can see when you are done. I have a clear elbow that allows me to flush so its useful to see how the flush is working as the water becomes clear.
DW, 3 Kids and our Goldens
2012 Shamrock 233S
2008 Toyota Sequoia 5.7L 4WD