Either can be correct. If you are going right away, you say, "I am going." I am going to the store I am going down the street I am going to my room. In your example you would say, "I am going to the university" if you were about to go to class. If you are planning for the future you say, "I am going to go" What are you doing next year? "I am going to go to college.' When are you going to the store? "I am going to go tomorrow." Who are you going to the movie with? "I am going to go with Paul." In your example if you are leaving and someone says, "where are you going to go?" you answer "I'm going to go to the University."

1. What's the difference between "I'm going to college" and "I'm going to go to college"? ("I'm going") is present tense, that is - it is happening now. ("I'm going to go") is a future tense, therefore it will happen in the future. 2. Imagine I'm about to leave home and someone asks me "where are you going?" Is it correct if I say "I'm going to college/university." or I should use "go to" after "going to" and say "I'm going to go to college."? - - You are going to go to college (as in the future). If it was within a few hours, you would say, "I'm about to go to college" OR "I'm going to THE college."

Both are completely correct in that context. The I am going to go to college could imply you are doing something else before you go to college in that context but doesn't have to. The main difference between these two structures is if you are asked that question: "are you going to college?" By someone who doesn't know if you are a student or not. The two answers then imply diffrent things the first being yes I am a student. The second being no I am not a student right now but I will be in the future.