Gain enough fame to begin and complete "The Gobbiebag" series of quests to increase your inventory space.
To increase space in your Mog House, gain enough fame for the "Moogle" series of quests.
To increase space in your Storage, set up the appropriate furniture in your Mog House. Bookshelves are generally recommended as being the most efficient.
To increase space in your Mog Locker, talk to Fubruhn in Aht Urhgan Whitegate.