When it comes to basic maintenance for your copper kitchen sink, really all you have to do to clean it is to use a mild soap with warm water and rub it in with a soft cloth (which in this case means "nonabrasive"). Then rinse with clean water and dry with a clean soft cloth. Depending on how thick the copper in your sink is (also known as the "gauge"), you may have more or less leeway when it comes to rubbing. Very thin-gauge copper (the larger the number, the thinner the copper) is not only more likely to dent, but also has a tendency to be more delicate. Never use abrasive cleaners on your copper kitchen sink.
Generally, you will need to keep up with the dishes, because leaving something to soak for a long period of time can ruin your copper sink. You will also need to be sure to not let hard water sit in or dry onto your sink, since minerals can be deposited on your finish and can require abrasive cleaning methods for you to get off. If you ever have to put anything acidic in your sink (such as vinegar, lemon juice, or any other acidic fruit), be sure that you leave the water running. Rinse the sink out as soon as possible, and then wipe it dry with a soft cloth. Oh, and be very careful about dropping sharp objects in your sink. If cleanser and acid do bad things to it, imagine what scraping a knife around the bottom would do to your copper kitchen sink.
Now, depending on whether you want the patina (the slight discoloration that develops as copper ages) to accumulate or be preserved, or if you would rather that your sink to look shiny and new, you will have to maintain your sink in a different ways. If you want the patination (the process of the development of a patina on your copper kitchen sink) to be slowed so that your sink is shiny, you need to buff your sink with wax specifically designed for the process. Another possible method would be to use wax that is food-grade, which could be beeswax as well as a clean, soft cloth. You may find that you have to sand and/or polish your sink when it finally does start to age, so that the sink finish looks shiny again; this takes time and effort, and this process may not go so well if you have a sink that has a high gauge (i.e. is too thin). But some sinks come with patina already on them, in which case, the finish can be preserved by gentle cleansing, drying, and buffing with wax and a clean soft cloth. If the patina is not yet apparent and you would like for it to become so, simply use the basic cleansing method, don't wax it, and give it time.