Gold and silver can develop tarnish if they aren't stored inside a cloth bag. Bags also prevent chains from tangling with other jewelry pieces. For long, delicate chains, insert them into a clean plastic drinking straw then fasten the ends of the chains. No more tangling!
I now use a jewelry box that has little compartments and hooks. I can hang necklaces inside and separate rings and earrings. My jewelry is organized and easy to find. If you don't have a compartment style box, then use the aforementioned bags to keep items separated. Even plastic ziplock style bags are better than leaving items loose to become tangled or lost.
When traveling, pack the items in separate baggies and bring them in your purse or carryon. If you stick the jewelry in your checked luggage you increase the chances of it becoming lost or stolen. If you are bringing expensive jewelry or that with a high sentimental value, keep it on your person at all times or store it in the hotel safe when you aren't in your room.