Prior to his uncle's departure for Spain in 1386, there's no question that Richard II was deeply suspicious of John of Gaunt, not least because various people were in the habit of telling him that Gaunt had designs on his throne. They were probably both glad to see the back of each other!
The events of 1386-1388 certainly gave Richard cause to think again. When Gaunt returned in late 1389, he was warmly received and over the next few months was appointed to the council, had the palatine powers he enjoyed in Lancashire extended to his heirs, and was made Duke of Aquitaine. To the annoyance of some, King Richard often walked arm-in-arm with his uncle, and on occasions even wore his livery collar of esses - 'as a sign of the complete amity between us,' Richard explained, probably with a smug grin on his face.
Though the Appellants, particularly Gloucester, were brought in from the cold, there is little doubt that he had not forgiven them, and this shows in small ways. Gaunt's heir, Henry Bolingbroke, received very little from the King in the way of offices or appointments, in complete contrast to the lavish appointments rained on Edward, Earl of Rutland, his junior in both precedence and age.
The next few years were relatively free from discord, and even Richard's biggest critics struggle to find fault with his performance, though they generally give the credit to Gaunt and the Queen for their moderating influences. Of course, it was rather difficult for the opposition to make many waves as long as the King and Gaunt stuck together. Gaunt was enormously wealthy - by comparison Warwick the Kingmaker was a pauper - and, as part of his attempt to take over Spain, he had retained vast numbers of the English knighthood and gentry in his service. These factors were eventually to rebound on Richard to some tune, but for the time being, they were his life insurance.
As an aside, some of you may enjoy reading this review of Alison Weir's book on Katherine Swynford which comments on the private side of Gaunt's life.
Also there's another royal geneology site here for anyone still struggling with who is who. It's very comprehensive.