Got a new phone? Your routes can come with you, but how depends on how you moved.
Android to Android, same Google account
Restore your Google backup while setting up the new phone. When you open Runplanner, tap Restore in the Welcome back message. This only works if the old phone had a screen lock. See Automatic phone backup.
iPhone to iPhone
Set up the new iPhone from an iCloud backup (or move everything with Quick Start). Your routes come along with the app.
Android to iPhone, or iPhone to Android
Make a backup file on the old phone and restore it on the new one. See Back up and restore with a file.
Already set up the new phone without a backup?
If the old phone still works, make a backup file there and restore it on the new phone. Restoring only adds what's missing, so it won't remove routes you've already made on the new phone.
Routes are stored on your phone, not in an account, so there's no login to recover them with. Save a backup file every now and then to be safe.