The modern subdivisions you mention were actually already in place by the American Revolution. The biggest operational differences would the level of autonomy granted to each unit, which is tied in part to the comparatively lower amount of firepower these units had compared to modern formations.

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Chapter 4 of Regulations for the Order and Discipline of the Troops of the United States (page 14 of the linked PDF), better known as Von Steuben's "blue book", lays out the formation and organization of a regiment, which was the basic combat unit of the time. This book was the operational manual of the Continental Army, containing the description of how the army was organized from the largest to smallest level, the technicalities of loading, firing, and marching, and a guide for the conduct of officers, NCOs, and private soldiers.

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The smallest unit described by von Steuben is the file. In the two-rank formations used by the Americans and British during the Revolution, a file was the two people who stood one behind the other when the regiment was formed into line. Even when marching in column, it was important for file partners to move together, particularly when the column needed to rapidly deploy into line-of-battle^(1.) Four files together made a section^(2). Two sections, in turn, made a platoon, and two platoons made a company. American regiments by the end of the war consisted of nine companies, though the ninth was organized as light infantry and combined with other light companies to make their own battalions on campaign. Regiments of more than 160 files were divided into two battalions, which would still usually operate together. 3-5 regiments made a brigade, and multiple brigades were formed into divisions. Note that these tended to be a bit more ad-hoc than we'd think of an army division being today, and they were generally named after the major general who commanded them. The corps as an operational unit would not emerge until the French Revolution.

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As mentioned before, the regiment or battalion was the major operational unit of the day. Companies were sometimes detached for garrison or fatigue duty, but in battle companies were supposed to stand shoulder-to-shoulder in line-of-battle. Sometimes, the broken and wooded terrain of America made this impossible, and company commanders could exercise some degree of initiative.

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1. The lines (x-axis) of the formation were called ranks, and the y-axis pairs called files. Thus, the term "rank-and-file."
2. The Continental Army also issued (in theory) a tin cooking kettle to each section, which out of combat was referred to a mess. Pooling the resources of a mess helped me trade excess food (do you really need 8 pounds of flower for 8 men each day?) for things that the army would not or could not buy, most notably fresh vegetables and fruit.

Sources,

Martin and Lender, *A Respectable Army*

Spring, *With Zeal and with Bayonets Only*