Baking soda is the first solution for your dingy whites. Mix the baking soda and some water and soak the mixture on the whites. After that wash the whites. You will see the results.
Waiting whites with aspirin and hot water 7-8 hours can be a solution. The ingredients of aspirin have bleach effect on whites.
Everyone knows bleach effect of lemon on hair and body. It can be used on clothes too. Rub whites with lemon 5-10 min and then wash the clothes. You can also use white vinegar after using lemon scrub.
When you are in restaurant and spill something on your white clothes, you can use salt. Add some salt on the area and press it. After you come home, it will be easier washing dingy whites.
If you are looking for fast solution, then borax is the only solution. Add 1 tablespoon borax in your washing machine and see the results.