So we start off with Jeff, his widowed mother Ellen (Jan Clayton), and his grandfather George "Gramps" Miller (George Cleveland) on a small farm in an unnamed state. Usually Jeff or someone else got into trouble and Lassie would have to come to the rescue because that's what dogs do. Also hanging around the farm were also Jeff's buddy Sylvester "Porky" Brockway (Joey D. Viera, back when he was known as Donald Keeler) and his basset hound Pokey.
In the fourth season, George Cleveland died and both Clayton and Rettig wanted to leave the show. Besides the juvenile star was growing facial hair. So Gremps death was written into the script, the farm was sold, and Ellen and Jeff moved away to the big city (if I remember correctly). The big city was probably Capital City, the oft-mentioned county seat. And because they could not take Lassie to the city, she stayed behind on the farm when Paul and Ruth Martin (played first by Jon Shepodd and Cloris Leachman, then by Hugh Reilly and June Lockhart) and their foster kid Timmy, the one that never really fell down a well (Jon Provost). Lassie continued doing his doggy duty by saving Timmy for the next seven years.
The next cast switch took Lassie into the U.S. Forest Service for five or six years. Lassie then wandered around on her own for a season before ending up at the Holden Ranch for orphaned boys for the last two years of the series.
But let's head back to the beginning -- season 1, episode 1, a two-parter which set up the Jeff-Lassie relationship.
Episode 1, part 1, September 12, 1954:
And episode 1, part 2, also September 12 1954: