Companies that refurbish phones have access to various specialized chemicals and machines for renewing cruddy old phones.
Amateurs don't have to make a major investment, because most people already have two very effective chemicals in their bathroom.
Rubbing alcohol is almost a universal solvent, very effective at cleaning off ink, label adhesive, and all kinds of goo from phone bodies and handsets. It's also good for cleaning cords.
Toothpaste does a good job at polishing out minor scratches and dull spots. You can apply it with a toothbrush, rag, or fingertip.
If you want to give your phone a good shine, get out of the john and go to your garage, and get some car wax.
Use these chemicals sparingly, and don't get water on the phone guts.