What type of Visa should I apply for if I'm studying AND touring China?

Hello! I'm going to Shanghai for a two-week class in June this year. I know I should be applying for an F type, however, I'm planning on going to Beijing for a week to visit a friend, after I finish the class. Since I will be doing both studying and touring, which type should I apply for?

Also, I received an invitation letter from the University in Shanghai, but it says I will only be staying for the 2 week long course. Since I will be staying longer, do I need proof of where I will be staying or who I will be staying with after?
Since I do not know yet if I will be staying at my friend's or in a hotel, I'm not sure how I could validate this.

Thank you!