what is the difference between washing soda, baking soda, and dish washing soda?

Washing soda and baking soda have slightly different molecular structures, Na2CO3 (washing soda) vs. NaHCO3 (baking soda). Both are alkaline in nature.

From Wikipedia:

Washing Soda:

Sodium carbonate (also known as washing soda or soda ash), Na2CO3 is a sodium salt of carbonic acid. It most commonly occurs as a crystalline heptahydrate, which readily effloresces to form a white powder, the monohydrate. Sodium carbonate is domestically well known for its everyday use as a water softener. It has a cooling alkaline taste, and can be extracted from the ashes of many plants. It is synthetically produced in large quantities from table salt and limestone in a process known as the Solvay process.

Baking Soda:

Sodium bicarbonate or sodium hydrogen carbonate is the chemical compound with the formula NaHCO3. Sodium bicarbonate is a white solid that is crystalline but often appears as a fine powder. It has a slightly salty, alkaline taste resembling that of washing soda (sodium carbonate). It is a component of the mineral natron and is found dissolved in many mineral springs. The natural mineral form, nahcolite, is found in dissolved form in bile, where it serves to neutralize the acidity of the hydrochloric acid produced by the stomach, and is excreted into the duodenum of the small intestine via the bile duct. It is also produced artificially.

Since it has long been known and is widely used, the salt has many related names such as baking soda, bread soda, cooking soda, bicarbonate of soda. Colloquially, its name is shortened to sodium bicarb, bicarb soda, or simply bicarb.

To learn more about each substance, see the sources listed below.

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Washing Soda Wiki

In more practical terms:

Washing soda dissolved in water makes the washwater feel slimy. Dry washing soda is a soft powder and can't be used as a cleaning paste.

Baking soda dissolved in water doesn't change the feel of the water. Dry baking soda is gritty enough to use to scrub stains.

Dishwashing soda, I don't know.