Well, if it won't actually turn over, turn on the lights and watch them when you try. If they don't change, but you hear a click, then its probably a bad solenoid or starter. If you don't hear a click, then you have a bad connection from the ignition switch to the solenoid or a bad solenoid.
If you hear a click and the lights get dim, then frozen engine, or bad starter. If they go almost out, but nothing smokes, then weak battery even if it checks good.
If they go almost all the way out and you see smoke, then bad connection where the smoke is coming from, or shorted starter.
That should get you some answers.