1. If a fighter leaves a carrier is the sea zone the carrier is in counted as it's first move, or do you move the fighter to the next sea zone and that is its first move?
You only use a movement point when you cross a boundary between two zones, so the sea zone containing the carrier the fighter launched from doesn't count.
2. During movement phase, can any sea or air unit move back to the base board without costing any supplies? Also, i know you have to pay a supply to move a unit from the base board onto the game board during deploy phase, but if it is during the movement phase can you move sea and air units from the base without paying a supply fee to get onto the main board?
Yes to both. However, you may only move to or from the Base Card using one of the two "Route to Base" sea zones when moving during the Movement phase.