First, a couple of notes:
* Consider putting dirty dishes in a plastic tub, instead of the sink. This way the sink is always available for dish washing - or whatever else it's needed for.
* Do dishes after every meal or snack. The sooner you wash the dishes, the easier they are to wash. Besides, nobody likes to see a sink full of dishes waiting to be washed.
How to Wash Dishes by Hand
1. Scrape food off the plates and into the compost bin or garbage.
2. Sanitize the sink.It's one of the germiest spots in the kitchen. Soap and water work okay, but a little bleach or ammonia really gets things much cleaner. Sometimes I'll spray the sink with Windex (which contains ammonia), walk away for several minutes, then rinse.
3. Fill the sink with hot, soapy water. It's smart to put a rubber mat or plastic tub in the sink, to protect glasses and plates from breaking. Use the hottest water you can stand, but don't burn yourself. Rubber gloves make it possible to use hotter water - and protect your skin from drying out due to soap and hot water. By the way, don't fill the sink or tub all the way up, because the water level will rise once the dishes go in.
4. Start washing. There are at least two schools of thought on what to wash first. Some believe that things that touch the mouth (utensils and glasses) should be washed first, since the water will be hotter and cleaner. Others simply wash things from cleanest to dirtiest. Certainly pots and pans should be washed last, because they dirty the water quickly. Also, some people like to use a brush to clean dishes - others prefer a scrubby cloth or sponge. I like Scotch Bright scrub sponges because one side is rough but don't scratch surfaces. Ideally, whatever you use should be easy to disinfect. (For example, you can microwave sponges or wash cloths.)
To wash: Place the item in the hot, soapy water and scrub it while it's underwater. Lift up from the water to examine it. Scrub again, as needed.
5. Rinse with hot water as you go. If you have a double sink, run the rinse water in that. If not, just run it into the soapy water. Avoid letting the water run in between dishes, since this wastes water and money. If your dishes tend to look spotted after drying, fill a large bowl or tub with water - plus a splash of white vinegar; rinse the dishes in this. (Dump out and refresh as needed.)
6. Drain the tub or sink, if at any time the water seems too dirty. Refill with hot, soapy water.
7. Dry. There are two ways to deal with wet dishes. Some people place them on a dish drying rack; you may wish to place a rimmed tray (like a baking sheet) beneath it, to contain the water that drips off the dishes. Other people prefer to dry dishes as they go, using a good, cotton dish towel. (I find "flour sack" towels work best.) This method is less likely to leave dishes looking spotted.
Want more tips? Check out 10 Ways to Make Washing Dishes Less Miserable