I collect my pods when they start to split open. They are usually brown then or turning brown. I open the pods into labeled paper cups in order to allow the seeds to dry out. Once they are dry I store them in the freezer.
Because modern lily hybrids have diverse backgrounds with respect to the many species, there can be a lot of variation with seedlings. This is particularly true with Asiatic hybrids. A general rule of thumb is that the plant form is inherited from the pod parent (nice inflorescence, etc).
I plan on writing some FAQs but it will most likely be a winter project. In the meantime, feel free to ask questions here. There aren't many of us that do this and we sure do welcome you to our addiction!