There are a lot of ways to answer this question, but here are some previous threads to get you started:

It's 1856. I, a wealthy southern gentleman, want to send my younger son to the best university available. What are my options? What are my options if I don't want him associating with too many northerners? and

In the 19th Century, what were students doing when they were carrying out "recitations" in University? by /u/Borimi

How far back until HS becomes University? by /u/UrAccountabilibuddy

It's 1700. What sort of process could I expect for admission to some of colonial America's oldest colleges, like William & Mary, Harvard or the University of Pennsylvania? by /u/lord_mayor_of_reddit

These are from the FAQ section on Formal Education and Universities:

What was life like as a university student in 19th century Europe? by /u/chromopila, among others

What were the young men studying medicine at a university during 1700s and 1800s England actually taught? by /u/KMBlack, /u/hangmanhands, and /u/RedditBlueit