Between the time of the Quest of Erebor and the War of the Ring, Radagast the Brown seems to have been dwelling near the western eaves of Mirkwood south of the Old Ford and not far from Dol Guldur. His home was called Rhosgobel. He might have taken part in the White Council's assault on Dol Guldur that forced Sauron to withdraw to Mordor, but this is never made clear.
The local Woodmen might have come to Radagast for help and wise counsel. We know that he was especially interested in the beasts and birds, and that interest led him to neglect his mission in Middle-earth.
In the summer of T.A. 3018, Saruman recruited Radagast to seek out Gandalf and tell him to come to Isengard to receive the White Wizard's help. Radagast did as he was asked, not knowing that Saruman had been corrupted by the Shadow. This led to Saruman's downfall when Radagast later sent Gwaihir to Isengard with a message for Gandalf. The Eagle of course enabled Gandalf to escape from Saruman's tower.
We don't know what happened to Radagast after that. Elrond's scouts found Rhosgobel abandoned. Saruman might have murdered the Brown Wizard or had him assassinated. Radagast might have learned of Saruman's corruption and fled in to the South or East. I've speculated whether Radagast could have fallen into an enchanted sleep, only to wake up many centuries later and become known in later legend as the wizard Merlin.
Radagast was the one who informed Gandalf that Saruman wanted to see him. However, it's probable that Radagast didn't know that Saruman had been corrupted. It is possible that Radagast sent Gwaihir to rescue Gandalf from Orthanc.
Nobody knows what happened to Radagast after that. I believe he decided to stay in Middle-earth after the War of the Ring. While the Free People were more than capable of rebuilding their societies, the birds and beasts also suffered under Sauron. Radagast would probably help heal the hurts done to nature. I like to think he contacted the Valar, especially Yavanna, to get permission to go on this new mission.