Scott Hayward rebels against the plans and expectations of his father, Duster Hayward, a very rich oil tycoon. He drives to Florida in his red 1959 Chevrolet Corvette Stingray Racer to find himself. When he stops for gas and refreshments, he encounters Tom Wilson, who is on his way to take a job as a water-ski instructor at a Miami hotel. A chance remark by Tom gives Scott the idea to switch identities with Tom so he can learn how people react to him as an ordinary man rather than as a millionaire. Tom has fun staying at the same hotel and pretending he is rich.
Shortly after, Dianne Carter insists on taking a water-ski lesson minutes after Scott checks in with his new "employer." However, on the water, Dianne proves herself an expert skier, performing fancy maneuvers that gain the attention of the wealthy young playboy James J. Jamison III. Dianne confesses to Scott that she is a gold digger, assuming he is rich too. Scott agrees to help Dianne land Jamison, but ends up falling for her himself.
Scott persuades boat builder Sam Burton to allow him to rebuild Burton's Rawhide, a high-performance boat that was damaged when raced at high speed, and to drive it in the annual Orange Bowl Race, which Jamison has won the last three years. He sends for some "goop," an experimental coating one of his father's companies spent a lot of money trying—and failing—to perfect. Between his day job and working on the goop at night, Scott is run ragged, but he thinks he has fixed the goop's major flaw: losing its strength in water. With no time for testing before the race, he applies it to the boat's hull and hopes it will hold the Rawhide together. Duster learns where his son is and comes to see what he is doing. To Scott's surprise, his father is enormously proud of what he is doing.
Meanwhile, Jamison proposes to Dianne. Scott enters the suite before she can give an answer, but the playboy informs Scott they are getting married right after he wins the race. Dianne, however, decides to give up her scheme and return home.
In the race, Jamison takes the lead in his boat, the Scarlet Lady, but Scott passes him at the finish line. Scott then offers to give Dianne a lift. On the drive, he gives her an engagement ring he bought with the winnings from the race. Dianne insists that Scott take it back, but agrees to marry him. This prompts Scott to confess to Dianne who he really is. She does not believe him at first, but when he shows her his driver's license, she faints.
