You can obtain coins by completing puzzle levels and through events.
There are other ways to get them too, such as by purchasing them in the Shop, receiving them as rewards for raising a Doll's Faith level, etc.
Coins can be used for the following:
・Continuing a puzzle level after running out of moves
・Purchasing Mana Gifts
・Purchasing Magic Items
・Creating a team
・Changing your Player name
・Bringing Dolls home from trips immediately
Try to collect as many coins as you can and make use of them as you proceed through the game.