You can carry them over as long as the OS of your new device is the same as the one on your previous device. 
Take note that if the previous device and the new device use different OSes, you will not be able to carry your Gems and Coins over.

Candies are the currency you need to increase the EXP of a Toy.

When you place a Toy in town or on a stage, the Toy starts producing Coins. After a certain amount of time, the produced Coins can be collected. In other cases, Coins can be collected through rewards of various content and events. You can also purchase them using Gems in the Goods Store.

Tap the [+] next to the number of Gems in the upper right corner to enter the Gem Store (Goods Store). You can purchase Gems by selecting a Gem product in the Goods Store and making the payment.

Gems are used when purchasing certain items or using content.
For example, Gems can be used to purchase Toy Balls at the store, spin Gacha, or immediately complete the remaining time for Toy Ball openings or building upgrades.
Gems can be acquired through in-app purchases, in-game play, or event rewards.

* When you tap the Gem button or the button that displays the number of Gems to be consumed, the Gems will be used immediately. Please consider this before tapping the button.

Coins are the currency you need to purchase buildings or items.