There are several techniques for restoring tarnished copper and you may have to try more than one of them to determine which one is best for the job you are doing. If the build up of tarnish, or verdigris, is heavy, it will require a lot of work to remove all of it.
Step 1: Clean
Before you start to remove the tarnish on your item, wash it thoroughly with soap and water and dry it. This will help you to determine what is dirt and what is tarnish. In the case of a copper cooking pot or skillet, there may be some blackening of the bottom. This is not tarnish and can be scoured away with the steel wool and dishwashing liquid. You can also use commercial soap pads such as SOS. Use them in a circular motion to prevent leaving permanent marks.
Step 2: Remove Tarnish the Green Way
A green tarnish removal technique that has been around for generations is the use of lemon juice and salt. Lemon juice is acid based and will remove the tarnish. The salt acts as a scouring agent to help loosen it. Cut the lemon in half and dip it into the salt. Use this to scour the tarnish off the copper. You will need to redip the lemon into the salt occasionally. Rinse often to note your progress. You can also use bottled lemon juice if you are cleaning a large item as it would be less expensive than using lemons.
Step 3: Tarnish Removal Products
There are a couple of favorite tarnish removal products available for sale. A favorite of antique dealers and collectors is MAAS metal polish. This creme rubs on, dries then buffs off removing tarnish in the process. The tarnish will appear as a black substance on the rag you are using, so change rags often and wear rubber gloves as its hard to get off your hands, even with soap and water. You may want to wear a shop apron or old clothes also as the blackened debris will stain clothing.
This is true no matter which method you choose to use on your item. It will also allow you to buff the item to a nice copper patina. This product is widely available at retailers and online. A little goes a long way, so don't overuse the product.
Another good tarnish remover, especially on larger items that would demand a lot of time, is TarnX. This liquid is sponged on and works almost instantly. When the item is completely clean of tarnish, wash the entire piece in cold, clean water and buff to a lustre. If a bit of tarnish remains you can buff with the extra fine steel wool to remove it. You may want to use this product outdoors or with good ventilation as it has a very strong odor.