I have an entire application written in PHP files. Each file handles operations, database access and html printing by itself. Years have gone away and I learnt about MVC, frameworks and so on. I´m about to resell this app with some improvements, most of them visual, but we are talking of almost 70 files to edit!
Is there a way to, for instance, separate each file into controller and view, so I can edit just the view in a more clear way? Also, doing so will allow me to migrate to CodeIgniter.
Use a template engine, look at this options:
https://github.com/bobthecow/mustache.php
Or use a framework like codeigniter
PHP already is templating language. You should instead use what's there to create simple templates without any fancy 3rd party tools. This will let you separate you HTML from the rest of you code.
Also, I would recommend to separate the DB interaction from you applications core logic.
If you have a working application, then do not slap a framework on top of it. Frameworks are tools that improve initial development speed. But the app is already there. Now you will only get the drawbacks of a framework, without any benefits.
As for MVC, it is an architectural design pattern. Frameworks do not implement MVC. Your code does (or doesn't). To understand MVC you at first need good understanding of OOP principles and practices (SOLID, SoC, LoD, DI). Just because you add a framework, it will not magically "add mvc" to your code. It's not some kind of sauce.
Finally, I would strongly recommend for you to stay away from CodeIgniter and CakePHP. They are the two worst frameworks in PHP. They contain a lot of bad practices, such as: global state, php4 artifacts, computation in constructors, misnaming of patterns and breaking of the separating of concerns.
They do not implement MVC or any MVC-inspired design patter, instead they are just bad Rails rewrites for PHP.