I've been doing my dishes by hand for years and always do it the following way because it's easy and works well.
Step 1: Sort all dirty dishes in piles: stack all plates of the same size, stack all bowls, group all glassware, and so on.
Step 2: Fill the bottom of the sink with warm to hot water, add dish soap. (I used to buy whatever dish soap was the cheapest but have become a fan of Mrs. Meyers Lemon Verbena Dish Soap. It works well and smells great. I totally recommend finding a dish soap that you enjoy using.)
Step 3: Wash all the glassware first.
Step 4: Place all cutlery, spatulas, whiskers etc. in the water to soak them while you do the other dishes.
Step 4: Wash one pile of dishes after the other. I start with mugs and cups, followed by bowls and plates, then wash the cutlery that has been soaking, and end with the greasier pots, pans and baking sheets. Place a few items in the sink at a time and let them soak while you keep cleaning. To clean, wipe each piece with a sponge, use a coarse brush as needed (I love the Redecker Pot Brush), and rinse off each item with really hot water. Stack the cleaned pieces on a drying rack and let your dishes air dry.