You can take your Lugol's in a few ounces of water on an empty stomach. Eventually, you can consider adding salt to this same water with the Lugol's, but one thing at a time. As MissCast stated in a thread below, the water in your empty stomach won't stay long, and when you can get your breakfast put together, take your selenium and magnesium with food. I like to take supplements before I eat, but others prefer to take them after they eat.
In the doing of these things, you will figure out what works, including the time of day that works best for you.