Welcome to the confusing world of programming styles There have been lots of debates on MVC in PHP and the whole thing may look confusing, especially that the term MVC has been mostly misused in the PHP world being used to describe PAC or some other related pattern. Partly, this might be because MVC was meant for desktop applications and it needs some adaptation to the web.
I haven't formally studied programming patterns so I don't want to get into the debate what is MVC, PAC or some other acronym and which is better but I'd say any of those is much better than a non-structured application without any pattern. So it's a good idea to have a look at a few PHP frameworks, gather some opinions from others and decide which one you like best and try it out.
The thing to have in mind is that a framework will not do everything for you - it will get you basic skeleton for your application and handle most basic and used stuff like routing, error handling, secured access with login credentials, database access, templating, etc. This is often quite a lot for a start because it can save you a lot of boring coding the same thing again and again.
But what I'm getting at is that even when you start using a framework you need to learn good programming practices because in a large application still your own code will be the heart of what it does, not the framework. So it's a good idea to study good OOP concepts so that your code will be modular and not grow into a mess with time. One of concepts I found most useful was Dependency Injection.
What I also find useful is trying not to tie my code to the framework whenever possible. Obviously, some parts of my application will be tied to the framework but I try to keep that to the minimum. I attempt to write so that I can take my application (my controllers, models and views/templates) and move them to another framework without much effort. I imagine that in the future when some of my applications become really heavily used (!) and I need to scale - I may write my own very skinny framework and move everything there for performance reasons
Instead of getting confused it's important to do some research - but limited in time because this can last forever :). The funny thing I found is that the more experienced programmers are the more they are likely to disagree with one another - those with experience have strongest opinions on what is good or bad because they have experience - except every one is different so their perspective will differ. As an example you may scan this MVC vs PAC thread
As to what I use - most often Silex. I don't think there is any PHP framework that I particularly like, each one has some solutions that are over-engineered or I don't like for some reasons so I've picked a microframework because it doesn't get in my way too much. I also prefer plain SQL for database access rather than ORMs (which are offered by many frameworks) and this has worked pretty well for me. With time you will build your own preferences and programming style.
I don't think the folder structure is the most important thing - just try to keep files organized in logical groups and you will be fine. Some frameworks may impose some folder structure on you.
And the last thing - apart from frameworks learn and begin using Composer if you haven't started yet - it is a tremendous help in using existing libraries and also offers a very nice auto-loader for your classes - most often I enable psr-4 autoloading in Composer, which means the folder path is the namespace of the classes in that folder, for example file "App/Invoice/InvoiceIssuer.php" contains an
InvoiceIssuer class in the namespace
App\Invoice - this keeps things clean and organized. Composer is already used by most modern frameworks.