Stefan thought it'd be safer if they left Mystic Falls forever. They didn't tell anyone where they were really going, just picked up and left as soon as graduation was over.
Jenna was told they were going to Toronto for college together. Jeremy wasn't told much, just that she would be safe. Caroline and Bonnie didn't want to know anything—didn't want to be involved anymore—so Elena had said goodbye, told them she loved them, and knew she would never see them again.
For the both of them, Damon was the hardest person to leave. They both loved him and hated him equally, and they both knew he returned the sentiment. For this reason, they allowed him to tag along for a little bit, get him to believe that this course of action was the best way to keep Elena safe, and once he believed that, he left for who knows where. Once he was gone, they were alone in the world.
They lived in California and Wisconsin, North Dakota and New Jersey, New York and Texas. Spent a few weeks in Italy and England touring the cities and country sides. They traveled to wherever the wind took them, and they had time to laugh along the way. They were happy.
Elena became used to looking over her shoulder, making sure no one knew where they were, that they weren't being followed, that no one suspected anything was strange with them. At the same time, Elena got the chance to see the world, to pretend to be normal. There were no vampires—that they knew of—no werewolves and witches and curses. There was no one to threaten Elena's life.
They changed names sometimes, to keep the trail hard to follow. After two years or so, they even buried their past lives and started over, a clean slate.
Eventually Damon and Katherine, Caroline and Bonnie, even Jenna and Jeremy were all forgotten. They didn't matter anymore. And somewhere along the road, Elena let Stefan turn her. She knew that once Klaus found her, he'd kill everyone she loved, so she vowed to keep running until all her human loved ones were long dead.
Five hundred years later, and they moved back to Mystic Falls to see how everyone fared. They ran into Damon there and learned that he had been keeping tabs on them for the last fifty years.
They stayed in Mystic Falls for five years in the boarding house, the four of them. Elena and Stefan and Damon and Caroline, who Damon turned when she was twenty three. Apparently she was unable to stay away from the vampire allure that is Damon Salvatore and tracked him down in Maine seven months after he left them.
They all went back on the run together after they left Mystic Falls. Elena got one of her best friends back after being alone with Stefan for so many years, and Stefan got a brother. They were happy, but always fighting to keep it that way.
They were on the run, but they didn't really mind it. They knew their days were numbered. Klaus would find them eventually, and he would hate them just like he hated Katherine. He would kill them all. But they had time, and between the four of them, they knew they would last much longer than Katherine.