As long as your new device is on the same platform as your previous phone (e.g. iPhone and iPad both run iOS) and you are signed in to the app store with the same ID, you will be able to download any previously purchased apps to any new devices. The app is not transferable to a new operating system, so if you switch from iPhone to Android, for example, your previous purchases in the Apple App store will not be available for your new Android phone or vice versa. In that case, unfortunately, you would need to pay for the app again.
My app store doesn’t show I have purchased and downloaded it. What do I do?