Speaking as a professional (heh) who washed dishes six straight hours a night, six days a week, for four straight summers at a Taco Bell / KFC, there is a proper way to do dishes, and an improper way.
First: spray your dishes off with extremely hot water, trying to get all the major particles off.
Two: in a sink filled with soapy, scalding water (as hot as you can stand), put the sprayed-off dishes.
Three: let soak for as long as needed.
Four: take each dish from the hot, soapy water and scrub with a green scratch-pad in a side sink running straight cold water
Five: if you can't scrub all the crap off, return to hot, soapy water and grab another dish
Six: if you do scrub all the crap off, rise the dish under the cold running water and get it sparking clean
Seven: put in third sink (with water, doesn't matter what temp.) or on counter to air-dry
For things especially greasy, use the above method but let them soak in a super-hot (hotter than you can stand to touch), super-soapy sink all by themselves; if you mix them with other non-greasy dishes you'll just get the grease on the non-greasy dishes and make your job 1,000% harder.
They key is to spray out the dishes before hand and get as much crap free as possible; I had co-workers who would just throw dishes with food still in them into the sink and that eventually leads to just a swamp of water-logged food shit.
Also useful: cleaning chemicals; liquid ones can be okay, but I prefer the heavier, thicker cleaners, esp. if they have granules in them; power-based chems like Comet are also good, applied directly and mixed with small amounts of hot water.
The only problem with this method is the combination of near-scalding water and chemicals will destroy the skin on your hands; for the first summer of almost non-stop dish washing, at times I could barely clench my fingers because the skin had hardened so badly. And then layers of your skin will peel off, and you'll loose the upper layers of the dermis which insulate your fingers, and your skin will be so thin that the warmth from putting your hand on your cheek will feel like you're putting your hand on a stove. But eventually you'll get over all that.
So that's how you do dishes.