I'm currently writing a motivation letter for a certain English class and I want to say that I'm looking forward to graduating and going to an English-speaking university. Is that the correct term to use? Going to a university?
The phrase 'going to a University' is correct but it is informal. You would use this verb when talking to someone casually. If you're writing a formal motivation letter there are better verbs to use.
'Attending an English speaking University', as @lambie suggests, is more formal and would have been common several decades ago
'Studying at an English speaking University' - studying means to be a student
'Matriculating at an English speaking university' (@mkennedy) would be the way the University formally talks about enrolling there as a student. Strictly speaking, matriculation is the event of being admitted and starting study. It only applies to the first few days of attendance, or to the formal ceremony of admittance if there is one (@David Siegel).