First remember that for many readers "Colonel" and "Sergeant" are just as alien as "Oberst" and "Feldwebel". A civilian who doesn't know rank may well assume both are just some kind of soldier. To truly understand the rank dynamic, show how the two interact, or how one person reacts when he is preparing to meet (or surprised by) someone of another rank.
How do they think about each other? Sergeants worry about whether the colonel who is coming is a good or bad officer; Colonels don't have time to worry about the sergeant they are about to meet. Lieutenants worry about themselves, and occasionally about the next officer up the chain, while Captains worry about their lieutenants.
How do they act around each other? Use the right verb to convey the tone and intent. A lieutenant may question a captain or correct a sergeant but may feel the need to instruct an airman. A Major, on the other hand, will not explain much of anything - he will order or direct and then probably leave, unless he's recently promoted and still falling back into the habits of a Captain.
Speaking is pretty simple because you have the honorifics and shortcuts that make military speech so efficient. But even the difference between "yessir" and "yes, sir!" lets your reader understand the speaker a little more.
Show the work that they have to do. A sergeant's job is greasy, muddy, back-breaking, or tedious. A lieutenant's job might be more accurately described as titchy, stressful, or never-ending. A colonel's job might not seem like work at all until you realize he's doing it from the time he wakes up until the time he sleeps, for months on end. Describe the tasks they are working on when the narrative interrupts them.
If you put enough clues about their clothing, work habits, and relative moods, the reader will understand the people and not just their relative ranks.