The purpose of the trip they were taking was for Earl to have a new start and to turn a fresh page in the book of his life. This desire to begin again is unfortunately not triggered by Earl's essential goodness to escape his past and to start afresh, but he is forced to leave Montana out of necessity. As a petty criminal, Earl's actions are catching up with him. He has tried to cash various bad cheques and now needs to flee Montana in order to escape the law and a prison sentence.
What is ironic about the road trip that Earl makes with his girlfriend and daughter is the way in which, although it starts so happily, it quickly turns into something of a nightmare for Earl, as the stolen car he is driving breaks down just as his relationship with his girlfriend breaks down. This in turn acts as a catalyst for the epiphany that Earl experiences when he reflects on the curious sense of powerlessness he feels in life:
There was always a gap between my plan and what happened, and I only responded to things as they came along and hoped I wouldn’t get in trouble.
Even though ironically the purpose of the trip was for Earl to seek his fortunes elsewhere, the trip becomes yet another reminder of how little control Earl seems to have of his own life.