I have been a fan of Heinlein for a long time. He has the feel for what an academy would be like when he details the Terran Navy in Space Cadet; and when he details basic training and OCS for Starship Troopers.
In both cases there is usually some sort of ground installation where you can have games, parades and other big outdoorsy type of activities. To look for a common example just look to the USMA, USNA or the USAFA
http://www.usna.edu/
http://www.usma.edu/
http://www.usafa.af.mil/
For an international feel I also include
Sandhurst, the British academy
http://www.atra.mod.uk/atra/rmas/
The French Equivalent is St. Cyr but a good site in english is tough to find. As well the Russians and Chinese have similar academies, but sites are hard to find.
A good general rule of thumb is that most academies are founded near to the center of government. This allows oversight as well as easy access to a supposedly loyal group of men and women in the event of a catastrophe, revolution or other bothersome event. (Unless it is a Cadet's coup you are dealing with!)
The central halls of military academies contain most of the book learnin' one would need, but ever since their inception (in the late 1700's-early 1800's with the invention of the national army on the French model and the creation of the general staff on the Prussian model) there have been some form of studies abroad. Whether it was as a midshipman aboard a frigate in Chesapeake Bay, with a home regiment in the Devonshire moors, or at the jungle training facilities (co-located with the ESA launch facilities, protected by the Legion Etranger) in French Guinea, the young cadets would always serve some time with active units doing the job of very-junior officers. Often this would mean something like sub-platoon leader, military secretary, general dogsbody or what have you. This was supposed to rub some of the spit and polish off of the young gentlebeings and show them what a modern military was like in action.
To go back to the Heinlein example (USNA '29) he had both of his protagonists go through an induction process, initial training dirt-side, then posting to an advanced facility for further study. While at the advanced facility they were posted to junior officer positions with serving units. In both cases they were sort of "third-lieutenants," no real command authority, but lots of responsibility.
In both cases something big came down during the assignment, and the heroes were able to learn what it truly meant to be "an officer and an gentleman."
I think that in the 56th century the model will likely be the same. So long as the academy in question truly wants to train effective officers instead of prettily dressed hangers on about court, then it will have an academic core curriculum, often heavily geared to practical or engineering type degrees, supplemented by specialist training and junior officer on-the-job-training at distant locales.
When all is done they will graduate in 3-4 years and have some sort of march, toss hats in air and get first assignments.
Then their long career will begin.
There is another aspect of the military academy tradition not often looked at, alumni associations. These tend to be very influential in both civilian and military life for academy graduates. They will look out for grads, help their careers along, and honor them for distinguished activities. As well they will hit them up for help and donations when the grads hit the big time. This is certainly a hook for characters that are academy grads. The "ring-knocker" tradition is real.
For those of you in the know…
"Will no one help the son of a widow?"
Or for the 3I…
"Will no one help a child of Arbellatra?"
Father Fletch, M.Ed., Ph.D., DD
P.U.C.L.Y.E.D.