You can say all three, but they mean different things.
"Getting into" refers to when the university accepts the student. In the U.S., this is usually in the spring before their studies will begin.
"Entering" refers to when your studies start. In the U.S., this is usually in late August or early September.
"Going to" refers to the entire time you are there, so "after going to" means after you leave (with or without graduating).
So, using "getting into" would mean they neglect their studies in the last few months of secondary school. That is possible.
Using "entering" would mean they neglect their studies at the university. That is also possible, but it is different.
"Going to" would mean they neglect their studies after they leave the university. That would not make sense.