Hi, diana. Cedar is a very pretty wood with a lot of color variations. It is usually just given a clear finish or unfinished but since it is so soft, I think that a finish gives some protection. You do not want to finish the inside since that will block the smell and the moth protection.
The way that I would go about it is to first apply a wash coat of shellac to the outside. The wash coat is made up of 1 part white shellac to 5 parts denatured alcohol. The can of shellac will be date stamped and buy the freshest that you can find. Just put it on with a foam brush but since the wash coat is real thin, be careful that you don't get any on the inside. Now sand very lightly with 150 grit paper both inside and outside, vacuum and tack rag.
Then I would apply a coat of oil based, satin polyurethane to the outside only. Follow the instructions on the polyurethane can for applying a second coat. Two coats of poly should be enough and give you some protection from wear, tear and dirt.