1) Do not "preheat" copper pots
Since copper conducts heat so well, it heats up very quickly. Preheating and other "dry heat" tasks (like toasting rice) should be avoided. The sensitive tin lining can melt if overheated. Avoid "dry heat".
2) Avoid scouring
Never scour the tin lining of your copper pans (with scouring pads or steel wool for example.) This will cause the soft tin lining to wear it away much faster than normal. To clean cooked-on foods, fill the copper pan with water and a bit of dish soap, and simmer for 15 minutes while you're cleaning up the rest of the kitchen. To accelerate the process, you can also use a scraper made of bamboo or other wood which won't harm the tin.
3) Don't sear in copper pans
The tin lining of a copper pan melts at only about 450 degrees F. To sear meats at high heat, choose cast iron, aluminum, or stainless steel instead of your copper cookware. Browning ground beef or a chicken breast is fine, but searing a filet mignon or a piece of tuna at high heat is not.
4) Use wooden or silicone utensils with copper pots
Avoid scratching the soft tin cooking surface with steel utensils.
5) Polishing 101
There are definitely two schools of thought here. Some love the bright look of polished copper - some love the soft tarnished "penny copper" look. Do YOUR thing; The copper will do ITS thing either way. If you are in the polishing camp - here is a great food-based polish:
Dissolve 1 T salt in 1/2 cup of white vinegar, and add enough flour to make a thin paste (like Elmer's Glue). After washing the pan, dip a moist paper towel in the polish, wipe on the copper for about 30sec, then wash off with warm, soapy & water. Dry WELL to prevent water spots.
If you do this after each use it will take about 30 seconds of extra time. Waiting six months and then polishing will require a lot more effort!