Should I go to a private college like PES/VIT or a newer IIIT (Guwahati, Kota, Sri City, Kurnool, etc.) for CSE?

Your answer entirely depends upon what your requirements are. 

• Two essential things that you would require as a CSE student most of the time are a good internet connection and a laptop. 
• Another truth is that just like any other college, you would have to surf YouTube to learn new stuff. This has nothing to do with the faculties. Nevertheless, the quality of teaching in institutes of national importance like IIITs is far better than in private institutes. 
• As far as the new IIITs are concerned, they would ensure you are given the finest of education along with decent placement packages. But at these institutes, you will not have many facilities for extra-curricular. 

Private colleges do offer good infrastructure and campus life but the teacher to student ratio, the quality of education generally do not compensate for the huge amount they charge. But if you look at IIITs, the student-to-teacher ratio is fine. 

Private colleges are always a gold option for the "fun" factor. But if you are looking for fine education quality, IIITs should be your first priority.

Just a side note, this doesn't mean that IIITs won't include the "fun" factor you're looking for. It simply won't have that great of an experience overall. Plus the infrastructure would not be very impressive either. But the quality of education you would find here would be unmatched.