Should I add the word "the" before mentioning a university?
I am a student at the University of ...
I am a student at University of ....
You'd use "the University" if you then followed it by the name. For example,
I'm a student at the University of Cambridge.
However, this would usually be said as:
I'm a student at Cambridge University.
Some universities do always take "the" though, for example UEA.
"I'm a student at University" just means you're studying at a university but it could be any one.
"I'm a student at the University" would only be used in specific circumstances where the university does things other than teach.
For example, Cambridge University is home to many research labs, and those scientists would say they "worked at the University".
"I'm a student at the University" would distinguish between being a student and working on the campus but not actually as part of the university.