I recently used the Slim framework to create an API. I want to have things clear to me.
- What exactly is the 
Slimobject? - What exactly makes 
$app->run()call the routing or the methods? - Is the RESTful interface itself handled by the 
Slim()object $app? 
Thank you in advance.
                        
The creator of the Slim Framework (Josh Lockhart) has a blog on the subject where he briefly explains some of the points you mention:
Calendar with slim
I personally learn by doing. There's a channel called, phpacademy on youtube where Alex and Co. have some great tutorials.
Blog
Authentication system