When and where does the story mostly take place?

Where the Red Fern Grows opens with an adult Billy Colman living in Idaho in the 1960s. On the way home from work one day, he rescues an injured redbone hound, reminding him of Old Dan and Little Ann, the coonhounds he had as a child. This encounter sends Billy down memory lane, and the rest of the novel takes place in the 1920s, when he was a child growing up in the Ozark Mountains.
Billy was raised on a farm in northeastern Oklahoma. The farm was nestled in a valley in the Ozark Mountains, isolating the Colman family from town but surrounding them with the wild beauty of nature. Billy describes the area with loving detail:

Behind our house one could see miles and miles of the mighty Ozarks. In the spring the aromatic scent of wild flowers, redbuds, papaws, and dogwoods, drifting on the wind currents, spread over the valley and around our home.
Below our fields, twisting and winding, ran the clear blue waters of the Illinois River. The banks were cool and shady. The rich bottom land near the river was studded with tall sycamores, birches, and box elders.
To a ten-year-old country boy it was the most beautiful place in the whole wide world. (chapter 2)

Billy spends much of his time at one with the woods, racing through the trees and streams with Old Dan and Little Ann. When the family is finally able to afford to buy a house in town, Billy wants to stay with his grandfather, and his parents understand—they know Billy doesn't want to be separated from his grandfather, his dogs, or the land he loves.
It's only after the coonhounds die unexpectedly that Billy moves with his family to town, where there are more modern conveniences and where he can receive a proper education. He never returns to the Ozarks, but as the adult Billy concludes the narration of the novel, he says that he would like to visit again someday.