Edward was sympathetic to some of the barons' desires for reform; at least, he sided with them for a time, possibly just looking to accelerate his accession to the throne. Henry prevailed against them, however, and his statements at the time show that he felt Edward had come under bad influence, and father and son were eventually reconciled.
When Simon de Montfort, 6th Earl of Leicester, led the barons in open rebellion, the Second Barons' War* (1264 - 1267) saw father and son working together. The barons wanted a council of barons to make decisions, not the king's favorites; not an awful idea, and Montfort did intend to broaden Parliament to include commoners, but their other "needs" were questionable. For one thing, Montfort's sons and supporters massacred hundreds of Jews in Worcester, Winchester, Lincoln, Cambridge, and Canterbury in order to eliminate debts owed to them.
Grievances against Henry were not without merit, given his increasing demand for taxes. Some of these demands had nothing to do with running England: for instance, he needed funds to attack Sicily on behalf of Pope Innocent IV.
Reformers versus royalists met at the Battle of Lewes in May 1264, at which Henry III was captured by Montfort's forces when Edward left his father's side to pursue some retreaters. Montfort took charge of government for about a year, but his governmental changes did not sit well with all of his followers: the nobles with him did not approve of his attempt to give power to commoners in Parliament. Loyalties shifted, and a year after Lewes, Edward's now superior forces defeated and killed Montfort at the Battle of Evesham.
Edward acquitted himself well as a leader of the royal forces to win his father's freedom, and although his earlier empathy with the reformers and Montfort could easily have led him to accept Montfort's reforms and become the next king (although with less executive and legislative power), he stayed true to his father's rule.
With order restored and the relationship between father and son on firm footing, it was time for Edward to prove himself in other ways. When he was 29 years old, he pledged to go on Crusade. This Ninth Crusade (1271 - 1272, sometimes called "Lord Edward's Crusade") is known not only as an extension of the Eighth Crusade, but also as the last Crusade ever actually to reach the Holy Land. But that's a topic for next time.
*The First Barons' War was alluded to here.