Busch was born in Las Vegas, Nevada. His first driving lessons came at the age of six when he drove around the cul-de-sac of his family's Las Vegas neighborhood in a makeshift go-kart. Although he could not reach the throttle, Busch still was able to pick up the basics from his father Tom, who controlled the gas pedal as Busch drove the vehicle. Busch worked in the family garage with his father and older brother Kurt as he grew, becoming crew chief for his brother's dwarf car team at age ten.
Busch began his driving career in 1998, shortly after his 13th birthday; from 1999 through 2001, Busch won over 65 races in legends car racing, winning two track championships at the Las Vegas Motor Speedway Bullring short-track, before moving to late models; Busch scored ten victories in late model competition at the Bullring during the 2001 season.
At the age of 16, Busch began competing in the NASCAR Craftsman Truck Series (now Camping World Truck Series), driving the No. 99 Ford for Roush Racing as a replacement for Nathan Haseleu, who was released midway in the 2001 season. He made his debut at Indianapolis Raceway Park, posting a 9th-place finish in his first race in the series. In his second race at Chicago Motor Speedway, he was leading until his truck ran out of fuel with 12 laps to go.
Busch was the fastest in practice for a 2001 Craftsman Truck Series race at California Speedway in Fontana, CA, when he was informed he was not allowed to participate in events at the track, due to the fact that the CART FedEx Championship Series, running at the track the same weekend, had its race sponsored by Marlboro cigarettes Busch was decreed ineligible to compete due an interpretation of the Master Settlement Agreement of 1998, prohibiting people under 18 years of age in participating in events sponsored by tobacco companies; Tim Woods III replaced Busch in the No. 99 Ford for the event. Busch competed in a total of six races in the Truck Series in 2001, finishing ninth twice, at IRP and at Las Vegas Motor Speedway.
Six weeks after the incident, NASCAR imposed a minimum age of 18 years starting in 2002 to prevent incidents of the sort from happening again; this was partially because Winston was the series sponsor of its premier Winston Cup Series at the time. When the age requirements were put in place, Busch switched from NASCAR to the American Speed Association (ASA) National Tour, finishing 8th in points.
In 2002, Busch graduated a year early with honors from Durango High School in Las Vegas, Nevada to focus on his driving career. That same year, he made his debut in the ARCA RE/MAX Series at Lowe's Motor Speedway, finishing twelfth in the No. 22 Chevrolet for WP Motorsports.