Only 4 Moonstones exist in the game, and some are missable, so be warned!
The first Moonstone is obtained at the very beginning of the game. You will enter in a sword fight performance where you must press buttons that your fellow performer calls out. Once you complete the performance, you will receive a short review of your performance based on how many out 100 nobles you impressed, and whether or not you impressed the Queen. Repeat the performance until all 100 nobles and the Queen are impressed. Not long after this, you will find yourself in control of an armoured Knight. Talk to the Queen to receive the stone.
The second is encountered much later, after spending some time in a Dwarf village (Rally-ho!). Along the path at the exit of the village are 4 coloured stones. Be sure to interact with each of them, and place the stones in another place further down the path (but still in the same area). You will then obtain the second Moonstone.
The third moonstone is part of a trading quest, known as the Friendly Enemies Quest. On the Forgotten Continent it is possible to encounter a friendly Jabberwock, who will request an Emerald. If you are doing these quests in order, you should have received one from the previous Friendly Enemy. Upon giving the Jabberwock the Emerald, they will give you a Moonstone.
The fourth Moonstone is a reward as part of an investment quest. At the beginning of the game, a Moogle named Stiltzkin, wearing a leopard-like headband will go on a journey. When you meet him on his journey, he will offer 3 items at a set price. You must buy these items each time you encounter him. The locations are thusly; Burmecia, Cleyra, Fossil Roo, Conde Petite Mountain Path, Alexandria (Once you return), Oeilvert, Bran Bal, Alexandria (After you return with your nice new ship). When you encounter him for the final time, he will sell a Moonstone. Note, however, that if you already have 99 of one of the three items he is selling, he won't sell to you until you have space.