I know the army was split into 5 categories Velites in front, Hastati in square formations behind, Principes behind the Hastati and Triarii behind Principes. (Cavalry on flanks)
I know that the you can spilt the army 'parts' up, in terms of experience/skill, Triarii the most experienced, then Principes and Hastati as the youngest/least experienced. I also know that the Velites threw their spears at the enemy, and then withdrew into the space between the Hastati, but what happened after this? How did the rest of the army actually fight, after the Velites had retreated?
The system is called a 'maniple System' - it was created during the Samnite Wars to counter the Greek Phalanx formation and the fact that the phalanx is really poor in certain environments (such as mountains, very hilly areas, forests, ect.). What would happen was the youngest troops, the Hastati, would attack after a rain of javelins from the Velites (used to try and confuse the enemy before the ramming). If things were very positive for the Romans, then the Hastati would take out the opposing formation without needing assistance. However, things often would go awry for the young troops regularly, for obvious reasons, and this is why the Maniple system is regarded as a 'phalanx with joints'; The maniples in each line generally formed with a one-maniple space between each maniple and its neighbours, and the maniples in each of the forward lines covering the gaps in the line behind, so that retreating troops of the forward lines could withdraw without disrupting those behind them. So pretty much imagine the first line attacking, as the next two look on, waiting for the commander of that line to give the signal whether things were good and the Principes were not needed, or if they were. From there the Hastati would pull back, Principes would go in. This allowed for the surviving Hastati to recover if necessary, and if the battle was going extremely bad to the point where the Triarii could not hold against the enemy without help, then the Hastati and Principes would be forced to go back in after a short period of recovery. Roman's usually would try, as far as cavalry goes, to use their cavalry to fight fire with fire, and drive off the enemies cavalry before aiming directly for enemy infantry formations, for the most part. The Velites would continue harassment from behind the melee infantry lines throughout the battle, until if they ran out of munitions, they would be forced to join in if necessary.
Also I forgot to include but just in case you were curious, the Romans did use the Phalanx formations before they implemented the maniple system.