Well, everybody says they went to the Starfleet Academy. And everybody who is anybody seems to have done so at the one (and only?) facility at San Francisco, where they all met the mysterious groundskeeper Boothby.
Possibly the vast UFP sees fit to put all of its officer-hopefuls through one unifying experience, because diversity is not desired - it is something the UFP already has in annoying abundance, and uniformity is the one thing missing from the equation.
Whatever the situation in the Trek reality, not only do we never hear of an officer who would not have gone through the San Francisco facility, we even hear Dr. Bashir (admittely jokingly) suggest the enlisted man O'Brien hadn't been paying attention at Academy lectures! Perhaps even enlisted training is centered at the California facility, and while the studies aren't academic as such, the general name of the facility is used anyway? Or perhaps enlisted specialists do take some academic studies?
While Starfleet has facilities all across the Federation and beyond, none seem to concentrate on offering training as such. But in "Coming of Age" we see a facility that organizes entry exams for the Academy (supposedly in the rare underage supergenius category for the likes of Wesley), and it seems to be an uncommon service for a Starfleet facility to provide: the starship had to specifically go there apparently solely to let Wesley take the exam.
This touches upon the entry age question: we don't know whether it's uniform even for any given single species or not...
Civilian training might in contrast be extremely diverse. It doesn't seem as if all children would have to go through any sort of common education, and certainly not at a specified age: some youngsters are shown already studying calculus when the older Jake Sisko has yet to tackle basic algebra.
As for "wargames" organizations outside Starfleet, we once hear of the Federation Naval Patrol, which supposedly is a youngsters' organization for playing with old boats. But Tom Paris considered it as an alternative to going to Starfleet, not as a pathway of eventually getting to Starfleet.
Timo Saloniemi