So, basically, I've developed an application that gets the data from the database, processes it and then displays it on a webpage and now I need to integrate it with joomla. I've read about MVC components and it looks like there are 2 MVC's available: new and legacy. Plus there is F0F. There are almost no manuals on how to work with new MVC and the last stable F0F version is like 6 months old so I am kinda afraid that it wont be compatible with the new joomla version. So how should I proceed? Should I follow the official manual on joomla docs and use a legacy MVC, try a new MVC or use F0F?
Legacy MVC will be supported until EOL of Joomla 3 (at least). I would just go ahead and use that. You will find many tutorials for legacy MVC and you will probably get the most help for that here or in the forums. Also (almost) all core components are developed with legacy MVC so you have many examples at hand.
The new MVC is already usable but you will have to implement a lot of functionality by yourself. I think it's easier to start with legacy but if you want an example how to use the new MVC you can look at the code of com_install because it already has been refactored.
FOF can probably help you to develop your extensions easier. I didn't get a chance to use it yet but imho I would learn the legacy MVC first.
You should never used legacy for new development as they are due to be phased out but left in to allow developers to adapt from legacy. FOF is akeebas framework on framework.