Installing is the normal Play Store flow. There are two things to do once the app opens.
Install the Play Store version over it. If Play tells you the app can't be installed, uninstall the copy you have first, then install from Play.
Your messages are safe either way. Android stores your texts, not lessms, so your conversations will all still be there. Filter rules, macros, and themes live inside the app and are covered by Android's automatic backup — if you've customized a lot, it's worth a screenshot before you start.
Focus Edition comes with you. Your subscription is tied to your Google account, so signing in to Google Play on the new phone restores it automatically. There is nothing to transfer and no code to enter.
The FAQ covers filtering, macros, and themes. For anything else, ask on the discussion board.