Access to a Mog House is free from the very beginning. Just look up the Residential Area of your home nation map and walk in. A Moogle will be waiting for you. If you have questions, ask the guard just outside the zone-line to the Residential Area. Your Moogle will also provide information about your Mog House.
Each nation has a Reputation/Fame 1 quest that will allow you to exit your Mog House to any other Mog House exit in the nation.