I have the WBS variety and here's what I do.
I fill both sides of the kitchen sink with fairly hot water. The warmer the better as I find the birds tend to relax more that way. And ladies, I know - I know! I can already see you rolling your eyes.
Ugh, the kitchen sink???!!! Hey, a guy has to do what a guy has to do.
I also fill up the bath tub too. On the right side of the sink I squirt some Dawn Dishwashing liquid in it. The left side is just plain rinse water and so is the tub.
Now, here's the main thing I wanted to recommend. Mostly I use Tresomme Shampoo or Suave or some other really cheap shampoo and lather up the birds quite well. BUT after coming home from a show with a case of mites on some birds, I have resorted to also using Zodiac flea & tick shampoo as well. It's the same amount of Permethrin as Adams but cheaper. I particular wash the backend fluff area well with the Zodiac and let it sit on the birds for awhile. Then I also take a Permethrin solution I make up and put in a spray bottle and give the birds a few squirts on the behind when I put them in the show cage and again when I put them back in their pens at home.
I take a soapy toothbrush, as Mike said, to their beards and muffs - and toes and feet if needed. Then I dunk them a few times in the water on the left side and then carry them to the tub where I dunk them a few more times and leave them to sit and soak their feet until I can pull them out and towel them off.
Lastly, depending on when I get all this done, I either partially or fully blow dry them or I put them in a pet carrier until I unload them at the show.
I should say that's probably more elaborate than many if not most. I know several folks who just have a few five gallon buckets. One with soapy water and one or two with clean rinse water. All they do is dunk the birds a few times in each and call it done.
Hope that helps.
God Bless,