Dear new2fly,
If your fishing vest doesn't have a fly drying patch you can pick one up for a couple of dollars at any fly shop.
I use one made out of foam, and when I change flies I put the used fly on the foam to dry out. At the end of the fishing session I return that fly to my fly box.
Truth be told, that is the way it is supposed to work, but I rarely am that organized.
Regards,
Tim Murphy