Velites - The poorest citizen class allowed to serve was known by several names throughout Roman history, yet each played a similar part. The Velites (Ferentarii, Procubitores, or Leves in earlier times) were not organized as their own particular units but were attached to the Hastati Maniples.
These units were lightly armed skirmishers who were deployed en masse on the battlefield in front of the heavier infantry lines, but moved back beyond the Hastati after discharging their javelins as the enemy approached. The Velites were also an important part of defense against war elephants. Using their greater mobility, due to lighter armament, Velites could avoid elephant charges and flank them. Spearing the elephants in the sides as they passed was a proven tactic.
The Velites were armed with the light javelin, or hasta velitaris, hence the name, and a gladius similar to that of their heavier infantry counterparts. Much like the Pilum, or the heavier javelin used by the main infantry, the velitaris was equipped with a finely sharpened and thin point making it bend upon impact, thereby reducing its ability to be re-issued in retaliation by the enemy. Their armor consisted of a light headpiece, generally covered by a wolf hide or similar animal. For body armor, they wore what they could afford, likely simple bronze chest plates or similar materials if anything at all. Additionally, the Velites were equipped with a small buckler or parma, for use in melee combat, though it was preferred that the Velites leave such action to their heavier armed counterparts.
Hastati - These were the first line of the Roman heavy infantry and were made up of the relatively young (inexperienced), but wealthier citizens. They wore a complete suit of defensive armor, consisting of the Legionary classic shield, or scutum, a simple bronze helmet (galea), a bronze breastplate or cuirass, and possibly leg greaves (ocrea). Those Hastati at the highest scale of wealth might have afforded better protection offered by chain breast mail, but it was likely uncommon for this class. Their weapons consisted of the classic legionary gear, including the pilum and gladius. Their helmets were adorned with purple and black feather plumes, ranging up to 18 inches in height. This was added to increase their apparent physical size and possibly appear more intimidating to the enemy.
The Hastati were the primary engaging force of the Republican legion and always met the enemy first in melee combat. If things went badly, or they simply needed to rest and regroup, the Hastati would roll behind the Princeps, and do so interchangeably until they were ready to re-engage.
Princeps - These were simply considered the elite of the Republican legion. They were in the prime of their careers for age and experience, and constituted the wealthier class of the citizen soldier. They were therefore the best equipped, though in a similar fashion to that of the Hastati. Aside from the likelihood of using scaled armor, or chain, rather than simple bronze breastplates, both weaponry and other armor was the same. It was simply the obligation of the Princeps to win the battle. First they would act in relief of the Hastati, but then would carry the fight in earnest. It was an interesting and successful tactic, wearing down the enemy with good, but not your best forces, than hitting with your best men right in the middle of the fight.
Triarii - Though the triarii were similarly armored as the Hastatii and Princeps, they played a much different role and were more akin to the Greek hoplites than Roman heavy infantry. The Triarii were usually the oldest and very experienced of the Roman army, and their job was to defend against disaster as a last resort, or to shock the enemy with a different look at the right moment. The Triarii were armed with the Roman equivalent of the phalanx style spear, the Hastae. If the heavy infantry were pushed back, the Triarii would charge forward with their spears, hopefully with the effect of shocking the enemy and allowing the Princeps and Hastati time to regroup. They were used as a last resort, and the Latin expression 'ad triarios redisse', or it has come to the triarii, became a general phrase meaning that something was in a desperate situation.
Cavalry - The Republican cavalry was designed for speed of movement, and were similarly equipped to the Hastati. Their general duty was to provide scouting and potential fast moving support to weakening parts of a battle line. At times the cavalry could be used to turn a flank or deliver a shock at the right moment, but their limited number generally made their capabilities limited. As the Romans were not considered great natural horsemen, the great effectiveness of cavalry in the Roman army didn't come until the inclusion of various auxilia cavalry like that of the Gauls and Germanics.
Prior to the great wars against Carthage, Roman strategy, and their enemy counterparts, was relatively simple. Most battles consisted of a straight march against one another, with little though of complicating matters. As most armies were levied and used on a need basis, initial and continued training was limited. Until the Punic Wars, Rome rarely maintained any semblance of a continual standing army, and it was just too risky to attempt many complex maneuvers with inadequately trained or experienced troops.
The wars with Pyrrhus of Epirus, and of course those with Carthage changed everything entirely, and the Roman battle strategies were forced to adapt. Years of brutal losses to Hannibal can be partially attributed to this fact, and of course to Hannibal's own brilliance on the battlefield. Until the rise of Scipio Africanus, there were few Roman commanders who implemented a great deal of strategy other than a straight frontal assault on an enemy position.