It depends who you're talking about. The Greek fighting style generally had the best fighters at the front, simply because that's where you got the glory from. The Romans followed the Greek Phalanx style until the early 300s BC, when they came up against the Samnites in the hills of Italy, who were able to run rings around the slow and unwieldy Phalanx.
They changed to the Maniple system, which split the main troops into 3 ranges - the New recruits, generally from 16 to 20, who fought in the front line, 20 to 40 year olds, who were the first reserve and finally the over 40 year olds, who were the final line of defence. They formed into smaller units than the Greek Phalanx, which was basically just one big pack of soldiers, which meant that they could be moved easier to flank an opposing force or reinforce a faltering line. The idea of the different ranks was that the younger soldiers would have more experience behind them to reinforce, as they were the most likely to flee, rather than at the back like the Greek Phalanx which allowed them an easier way of escaping a battle. Because most of the enemies of Rome would fight in a big bunch or expend a lot of energy on a charge at the beginning of the battle, having the more experienced troops at the back allowed them to take advantage of a flagging enemy who had expended a lot in the initial part of the battle.
They were able to build this reserve of veterans because the Maniple system was so successful for 200 years. It was more organised, disciplined, better equipped and led than pretty much all armies they faced at the time. The only times the Roman army would lose using this system was when they went away from their standard system of battle (see Cannae, where the Roman general clumped his army together rather than keeping gaps between the maniples, which meant that when Hannibal started to surround the Roman army, they had no room to maneuver and were slaughtered) or they were led into a trap (see Lake Trasimene, again against Hannibal, where the Carthaginian trapped the army against the lake, again leaving them with no room to maneuver.)
The Maniple system only collapses with the rise of large estates being run by slaves, which meant that their were fewer citizen soldiers who owned enough land to be eligible for military service, leading to the Marian reforms, which occurred around 100 BC and the professional army that led to the Empire. Soldiers of different experiences were mixed together, to allow for better organisation, training and motivation. A lot of the basic fighting tactics remained the same to the earlier system.