Summarize:

If it's really slow-draining, this may take a while, but if you remove the water, your drain-clearing mixture will be able to unclog faster. You have several options for creating a non-commercial drain opener. Most involve vinegar and another substance that create a chemical reaction when combined. See if you have any of these drain-opening agents on hand:  Vinegar (white or apple cider vinegar work) is the acidic base for creating the foaming reaction. Lemon juice is acidic like vinegar, but smells refreshing. This makes lemon juice a good option for clearing out clogged kitchen sinks. Baking soda is frequently used as a multipurpose cleanser. Salt will help eat away at the clog. Borax is frequently used as a multipurpose cleanser. No mixing is needed before pouring down the drain. The mixture will foam up on its own as the chemical reaction occurs.  For a vinegar and baking soda combination: use 1/2 cup of baking soda and 1/2 cup of white vinegar. For a lemon juice and baking soda combination: use 1 cup of baking soda and 1 cup of lemon juice. For a salt, borax, and vinegar combination: use 1/4 cup of borax, 1/4 cup of salt, and 1/2 cup of vinegar.
Drain any water from the sink or tub. Gather household cleaning/kitchen items. Pour vinegar and another drain-opening agent down the drain.