After having been exposed to basketball for years through his father's head coaching position, Bryce played basketball as the point guard for Valparaiso High School in Valparaiso, Indiana. As he progressed through high school though, Drew developed a rapid heartbeat, which required three surgeries to repair. Despite this difficulty, he led his team to the state final game, and was named Indiana's Mr. Basketball of 1994. He was also named the Gatorade Indiana Player of the Year his senior season in high school after guiding his team to a 28–1 season with the only loss coming in the state finals in overtime to the South Bend Clay High School Colonials.
Though recruited by dozens of schools, Drew eventually decided to attend Valparaiso University, then a member of the Mid-Continent Conference, for men's basketball. In his four years playing, Drew collected dozens of honors and records, including being ranked in the top 15 nationally in 3-point field goal and free throw percentage and leading the team to three consecutive conference regular season and tournament championships. He also collected three conference tournament MVP awards, two conference MVP awards, and is Valparaiso's all time 3-point field goal, and assist leader. He is also second in points for Valparaiso, being passed by Alec Peters in 2017.
During the 1998 NCAA Men's Division I Basketball Tournament, 13-seed Valparaiso was facing 4-seed Ole Miss in the first round. Valparaiso was down 69–67 with 4.1 seconds remaining in the game and Mississippi's Ansu Sesay at the free throw line. After Sesay missed both shots, the Crusaders came up with possession 94 feet (29 m) from their basket, and 2.5 seconds remaining in the game. On the inbound, the Crusaders used a play known as "Pacer". Jamie Sykes inbounded to Bill Jenkins, who passed the ball to Bryce Drew. Drew made a 23-foot 3-point shot, giving him his 22nd point of the night, and clinching the Crusaders' 70–69 upset and advancing them in the tournament. Drew proceeded to lead the defeat of 12-seeded Florida State 83–77 in overtime, with a 22-point game. Drew and the Crusaders fell to 8-seeded Rhode Island by a score of 74–68, with Drew scoring 18 points. Sports Illustrated would rank it the No. 5 sports moment of 1998, and Drew secured his place as a Valparaiso, Indiana, celebrity along with popcorn guru Orville Redenbacher.
Following his rise to fame in the tournament, Drew was selected as Valpo's first ever first round pick as the 16th selection of the 1998 NBA draft by the Houston Rockets. After playing with the Rockets for two years, Drew spent one season with the Chicago Bulls, and signed as a free agent for three seasons with the New Orleans Hornets (Charlotte Hornets during his first season with the team). Drew was then waived by the Hornets, and played professionally for the Valencia BC for a year.
In the summer of 2005, Drew was selected as the new assistant coach of the Valparaiso University men's basketball team. In 2006, Bryce was elevated to the position of associate coach, furthering speculation that he would eventually inherit the head coaching position upon his father's retirement. When Homer Drew retired in May 2011, Bryce Drew was hired as the head coach. Drew was also honored as one of Valparaiso University's 150 Most Influential Persons in the University's history.
On April 6, 2016, after five seasons as Valpo's coach, Drew was hired by Vanderbilt to be their head coach. In his introductory press conference, he stated, "No Vanderbilt team has ever made it to the Final Four, and we would like to be that first."
Drew is the brother-in-law of former University of Toledo and Philadelphia 76ers basketball player, Casey Shaw. Drew's sister Dana is Shaw's wife.
Drew's wife, formerly Tara Thibodeaux, was a professional cheerleader of the Atlanta Hawks. Tara is the daughter of former child actor Keith Thibodeaux who portrayed Ricky Ricardo, Jr. ("Little Ricky") on the TV series I Love Lucy.
His brother, Scott Drew, is the basketball coach at Baylor University in Waco, Texas.