The first shot of the movie is a car being cabled across a smoggy sky, and we soon learn that cars are super important to the Babbitt family—and really, the whole plot. (And the American dream, but we'll get to that.)
Charlie Babbitt is trying to make a killing selling Lamborghinis, but the EPA forbids him to turn them over to customers because they don't meet emissions standards. That whole situation creates some serious money problems for Charlie, and that's part of the reason he's so desperate for a piece of his dad's inheritance—and why he basically kidnaps his long-lost brother and holds him for ransom.
We also find out that Charlie has been totally into cars since he was a kid—in fact, a car was the reason he and his dad had a big falling out. His dad owned a car that Charlie really wanted to drive, and one day, he took it out for a spin without permission because he thought he had earned the privilege.
His father didn't agree, and when the police picked up Charlie, Father Babbitt wasn't super inclined to help Charlie out of the mess. Their relationship never recovered from the incident, and when Charlie's father died, he left Charlie the car… but basically nothing else. So, that car is a symbol of all the family angst and the breach between father and son.
Of course, we have to note that Raymond, too, was into cars—and unlike Charlie, Ray was allowed to drive that car all he wanted, apparently (at least, under tight supervision). Raymond is forever telling people "I'm an excellent driver"… and while that's not exactly true, Charlie (like his dad before him) finally lets Ray drive around a little bit to make him happy. So, it seems like Ray sees the permission to drive as an important sign of autonomy, kind of like Charlie did when he was a teenager.
Bottom line: Cars kind of become symbolically attached to the ideas of freedom and family in the movie, it seems.
But we would be remiss in our duties as symbol finders extraordinaire if we failed to mention the underlining motif of cars in a movie that's equal parts a) a roadtrip movie and b) a movie about American capitalism.
It's a car, guys. Arguably the most American of American symbols (besides all those eagles, baseball, and apple pies)… and completely, in a not-up-for-discussion way, the current reigning symbol of the American dream of freedom. But like a lot of movies that tackle the idea of the American dream of freedom, Rain Man uses the car-as-freedom as an ironic punch line. Because this roadtrip, in this coveted car, actually brings Charlie back to… the warm embrace of familial love.
And while that's a wonderful thing, it's also a very ensnaring thing. It's like the opposite of freedom, at least for a dude like Charlie. It's in his family car that Charlie awakes from his cold American dream and finds something much more real (and cozier).