I also have another question: Who would hold more rank? A cavalryman of private or an NCO of infantry? I'm aware that cavalrymen were almost always selected from the middle and uppermiddle classes (since the British and other nations could choose to do this), and prestige took a man a long way 200+ years back. This makes me curious as to whom would be more in command on the social scale.
This is a question with a few different answers, depending on which country you're asking about.
Special military academies did exist, broadly speaking. Napoleon himself attended the Ecole Militaire and graduated at the age of 16, at which point he gained a lieutenancy. Less than ten years later he was a brigadier general with command over the French revolutionary army's artillery in Italy.
French, Prussians, Russians, Spaniards, Brits, Dutch, Italians, and Ottomans all did things slightly differently. Can you specify a nation you're interested in?