On your new device, tap to open the App Store.
Usually found at the top right corner of the screen.
You'll see an option titled "Purchases". Tap on this to view a list of all apps that you have purchased (including free apps).
If you want to view apps not currently installed on this device, tap on "Not on this iPhone/iPad". This will show you all apps associated with your Apple ID that aren't currently on the device.
To download an app, just tap the cloud icon next to the app you wish to install.
This process is identical across devices like iPhones and iPads, provided they are logged in with the same Apple ID that was used to make the purchases. If you are switching to a device you haven't backed up, this method will still allow you to sync the apps you previously used.