This is the challenging part:
Short of seeing the actual source code I don’t see how anyone could really suggest alternatives without knowing how much you actually understand. You could consider posting a link to your source code repository assuming the code is yours to share.
Otherwise, consider just jumping in and creating a Symfony project and re-implementing some of your app. Don’t think of it as refactoring your app so much as a learning experience. You might very well end up tossing the entire project. But you will learn quite a bit about OOP in the process.
I recommend Symfony over some of the alternatives because, from an OOP standpoint, I’d say it is generally recognized as the best in the field and is well supported. But most of the other frameworks would be fine as well.