Yes, $15 should be the federal minimum, but somewhat higher may be necessary in the First World portions of the U.S.
Do you have ANY idea what a $15 minimum wage will do?
Yes - redistribute.
Of course I would back it up with a very generous welfare state to deal with the unemployment some imagine this would cause.
If nobody is working, where will the money for the welfare state come from, and where would the products on which to spend this money come from?
It won't cause much unemployment - instead it`ll cause a hell of a lot of inflation if done in a single step. Imagine most people to not be better or worse off afterwards.