During Robert's Rebellion, Aerys had died under suspicious circumstances, Rhaegar was coronated king when the war had ended.
Rhaegar would turn out to be one of the greatest kings ever seen. He was mercifull to Lord Tully and let him live with a small piece of land. He made the Royces the new lord paramounts of The Vale as they fought against the Arryns during the rebellion. Lord Jon Connington, squire and a dear friend to Rhaegar was given the Lord Paramouncy of Stormlands and be named Hand of the King. The North was given to the Boltons as they remained loyal to the crown aswell, but Rhaegar never really trusted them.
Under Rhaegar's rule, no rebellion ever happened. He succesfully reigned the throne for nearly 35 years, and kept the King's Peace together with the hand of the king, Lord Jon Connington. It is said that they passed away at the same time, a natural death.
Rhaenys Targaryen was the first daughter and child of Rhaegar. She inherited because her brother, the heir, was murdered in Dragonstone. Rhaegar had no other sons, so he decided that Rhaenys would inherit as the queen, by right. Rhaenys was quite interesting however. She had no remarks of a valyrian, but she was rather Dornish.
It is also rumoured that Rhaenys was in love with Princess Liane of Dorne. Not only that, it is also said that they had a threesome with Lord Willas Tyrell, who at that point was Hand of The Queen. Soon after, he declared a rebellion of an independance league.