Richard didn’t do very much at first, but at the end he was more heavily involved, and was basically just a pain in the ass for Henry II. He was only 15/16 years old at the time so he wasn’t a major player. Just to recap (and as you probably know, based on your question), the revolt started in 1173 when Richard, his mother, Eleanor of Aquitaine, and his elder brother Henry (“the Young King” since he had been crowned co-king with Henry II) and his younger brother Geoffrey openly rebelled against Henry II by allying with Louis VII of France (Eleanor’s first husband). Their other brother, John, was still a child at the time so he wasn’t really involved.

Richard’s first experience participating in the revolt was In July 1173, when Philip of Flanders, another one of Louis’s allies, besieged Drincourt in Normandy. It didn’t go well, and Philip gave up when his brother was killed in the fighting. Richard didn’t really do anything, he was just there, watching and learning.

When this siege failed Louis and Henry held a peace conference, and Henry offered Richard “half the revenues of Aquitaine and control of four castles; similar proposals were put to Henry [the Young King] and Geoffrey.” (Gillingham, pg. 48) Louis advised them to reject the offer, so the rebellion continued.

In November 1173, Henry II captured Eleanor:

“It may well have been the news of his mother’s capture which stirred Richard to take his first independent political action. Up to this point, still only sixteen years old, he had remained very much a background figure, drawn along in the wake of his elder brother, and both of them overshadowed by their protector, King Louis of France. But with his mother arrested it was now up to him to take charge of the rebellion in Poitou.” (Gillingham, pg. 48-49)

Richard tried to besiege La Rochelle but failed. La Rochelle was a big, rich trading port that depended on Henry’s support and Henry’s control over the trade routes in England, Normandy, and Aquitaine. It was a good strategy, but as an inexperienced teenager he had no way to execute it! So he established his headquarters further south in Saintes, but Henry II, who was still smarter and faster, outmanoeuvred him and took Saintes in 1174. Richard fled to a stronger position at Taillebourg. In July 1174 the rebellion was effectively over after Henry II defeated or captured everyone - except Richard.

Henry and Louis made peace in September but since Richard was still in rebellion, he was not included in the peace. Now that Henry was not distracted by Louis or any of Louis’ other allies, it was pretty easy for him to track down Richard:

“As his father approached, Richard retreated steadily, never once daring to stand his ground against him. Angered when he heard that Louis and the Young King had deserted him, the news none the less convinced him that the cause was lost. On September 23 he entered his father’s presence. Weeping, he threw himself flat on his face at Henry’s feet and begged forgiveness. His father raised him up and gave him the kiss of peace.” (Gillingham, pg. 50)

Richard accepted Henry’s peace offering (this time, half of the revenues of Aquitaine but only two unfortified castles). Richard made out pretty well, since Henry sent him to govern Aquitaine the next year - he was already Duke of Aquitaine in name, but now he was allowed to act as Duke in practice as well. So he ended up basically being rewarded for rebelling.

There are lots of biographies of Richard but the one I have used here is John Gillingham, Richard I (Yale University Press, 1999), chapter 4, “War Without Love, 1173-1174”, pg. 41-51.