I believe and have found in my experience and observation that example is the best method of instruction. As parents we should be willing and able to model the right behaviors and beliefs. If you are comfortable walking around the house nude, and letting your kids see you nude in the right and appropriate situations they will quickly learn and adjust to that. Where I had some difficulty was that I tried to change to a more naturist lifestyle after my kids were older. My youngest at the time were pre-teens and thus at the age that they would 'explore the boundaries' which caused some minor issues. My wife and I recognized this as the real issue and worked hard at not over reacting and using the situation to express the right behavior and timing. Overall my kids quickly adjusted and accepted the right standards and boundaries. It also helped with a couple of FHE lessons on our bodies/nudity and modesty as well as proper and acceptable touch.
My older kids took the opportunity to study, ponder and pray about the situation and thus come to their own decisions as to whether it was ok or not. Each one that did got the same answer which made all nudity issues much easier to deal with.
In my experience many parents are not nearly as direct and to the point with their kids and seem to think that they can't or shouldn't be so explicit and complete in their discussions. This tends to create ambiguity in what is presented and later problems. Be direct, support with scriptures and actual quotes from the prophets. Then discuss it with them, don't lecture. This way you ensure that they understand what is being said/taught.
I'll give an example of what I saw as a positive result of my teaching my kids. My daughter while attending seminary had a lesson on the Law of Chastity/modesty. During the lesson the teacher stated that "sex was bad and nasty" trying to indicate that sexual relations outside of marriage was wrong. My daughter was quick to speak up and stated that he was incorrect, that sex was good and awesome, just problematic outside of the bounds of marriage.
We teach our kids that modesty is the outward expression of an internal issue and I do not believe they have any lack of understanding as to what real modesty is versus what many at church seem to think it is.
RWB:
You must have some great youth leaders in your area. Unfortunately (especially here in Utah) the predominate teachings on the matter of modesty have little to do with the real definition of modesty and more to do with covering your body, it seems that a burkha might not be sufficient for many here.
"Modesty died when clothes were born."
---Mark Twain