Rome was the first state to have in its army a cadre of dedicated, experienced, and professional NCOs: The centurian. Roman (and Greek) armies had similar organizational set ups. The basic unit was the individual. Then next step up was the file. A file is the column of soldiers that face forward, so if a battle line was 100 men wide, it would be made up of 100 files. (Files were often 8 to 12 men deep.) Quite often, the most experienced soldier, or the best perhaps, would be the first one in the file. He would be the first one to engage the enemy. Behind him the men would be arranged by experience, less experience meant a man would be farther back in the file, with the exception that the second most experienced man would be in the rear of the file to offer encouragement (and probably keep the newbies from running in terror).
Files were arranged by experience as well, going from left to right. The most experienced guy in the formation would be in the front-right corner. This was because soldiers would nudge to the right to try to get protection from the shield belonging to the soldier to his right. As a result, formations would drift to the right. Putting the most experienced guy there would help avoid this.
In the Roman armies, this guy was the Centurian. He commanded his century, a number of files made up of about 100 men. He might be like a master seargent, or something like that. He was a career professional. IIRC, each file was commanded by the first guy in the file—eight-man units for much of Roman history—then the next step up was the centurian. As a career professional, he had a lot of experience, training, and knowledge, and with Rome’s proclivity for violence, he had plenty of opportunity to gain experience.
A lot of ancient armies were formed up ad hoc and led by an inexperienced aristocrat. What set Rome’s armies apart was that on the ground level there was a high density of quality leaders who could show soldiers what to do in everything from setting up tents to dealing with a barbarian axe-man. Because of this, the Roman armies could operate efficiently and be effective even in spite of poor generalship. IIRC, there are many instances where the experience and cool-headedness of the centurians saved Roman armies from disasters that would have caused other armies to collapse and be routed.
More prosaicly, I’ve seen the Roman weapons praised highly. The short sword forced the Roman soldier to get in close where longer weapons were ineffective. Additionally, stabbing is a quicker and more efficient action than slashing. A Roman soldier could protect himself behind his large, curved shield and then either pop up to stab, or reach around to stab. Since he made actual contact with the opponent, there was no need to actually see the other guy. One training exercise used a verticle pole securely anchored. The soldier would push against it with his shield keeping positioned to be well protected. Then he’d reach around his shield and carve up the pole. Since an opposing army in formation would force the opponent to stay fairly fixed, the soldier could neutralize longer weapons and attack without threatening himself too much. Note-worthy is the fact that, IIRC, Roman armies often did poorly in terrain that prevented regular formations. Fighting in forests in Britain and Germany went very poorly for them (IIRC, again).
A good, easy-to-read primary source is Roman Military Institutions written by Flavius Vegetius Renatus in three-hundred something-something b.c. There are lots of other easy to read books in the library. Just browse the shelves under military history and you’ll find plenty.