I started learning breakfalls some time just after I had taken 6th kyu. I think everyone is ready at a different rate depending on how well they take to the easier ukemi. In my dojo we will sometimes have a breakfall clinic after the first class has ended where anyone who would like to will be taught individually by sensei how to take them. We do not usually work on them as a class because not everyone is ready for that at the same time as we have people of all levels in the classes. After that it really becomes a matter of choosing to take that ukemi during regular training when the opportunity presents.
I did have some soreness in the initial attempts because of not knowing how to arrange my body properly and there is a sort of body shock that causes muscles to be really sore from all of the impact if you do a lot of them but eventually you learn how to land so that this is minimal.
At least that has been my experience.