Light is an oscillation of magnetic and electric fields. That's what the wave is, not a vibration in a medium. The second assumption, however, is against all common sense. This assumption violates common sense but has since been tested with equipment of such accuracy that if it wasn't valid, the discrepancy would have shown up. The first assumption actually means that you cannot measure absolute speed.
Speed is relative.
You can measure your speed relative to the Earth, the Earth's speed relative to the Sun, the Sun's speed relative to the centre of the Galaxy, etc. What you cannot measure is your speed absolutely. The second assumption means that many quantities that we thought of as constant and unchanging actually vary for different observers. These are the predictions that Einstein made. The mass of an object increases with its velocity.