Smith attended Kentucky Country Day School in Louisville, Kentucky. In 2013, as a senior, he hit.528 with 11 home runs and 36 RBIs along with pitching to a 7–1 record 0.87 ERA. He went undrafted in the 2013 MLB draft and enrolled at the University of Louisville where he played college baseball. In 2015, he played collegiate summer baseball with the Brewster Whitecaps of the Cape Cod Baseball League. In 2016, his junior season, he slashed.382/.480/.567 with seven home runs and 43 RBIs in 55 games. After his junior year, he was selected by the Los Angeles Dodgers in the first round of the 2016 Major League Baseball draft. Smith signed on July 17, 2016, for a $1.775 million signing bonus. Smith began his professional career with the Ogden Raptors of the Pioneer Baseball League and was then quickly promoted to the Class-A Great Lakes Loons of the Midwest League. He played in seven games for Ogden, 23 for Great Lakes and 25 for the Rancho Cucamonga Quakes of the California League, hitting a combined.246. He was named to the California League mid-season all-star team in 2017. He hit.232 in 72 games for the Quakes with 11 homers and 43 RBI. He was promoted to the Double-A Tulsa Drillers of the Texas League in July, but suffered a fractured hand after getting hit by a pitch in his debut with the Drillers and he spent the rest of the season on the disabled list. He split 2018 between Tulsa and the AAA Oklahoma City Dodgers of the Pacific Coast League, hitting.233 with 20 homers and 59 RBI. Smith began 2019 with Oklahoma City but was promoted to the majors for the first time on May 27, 2019. He made his major league debut the following day against the New York Mets and had two hits in four at-bats as the starting catcher. His first major league hit was a single off Steven Matz in the second inning. Smith's first MLB home run was a walk off against Héctor Neris of the Philadelphia Phillies on June 1, 2019. On June 23, 2019, Smith hit a walk-off three-run home run in the bottom of the ninth inning against the Colorado Rockies. It was the third straight game the Dodgers won on a walk-off home run by a rookie, an MLB record. He returned to the minors for a month and was selected to the Pacific Coast League team at the Triple-A All-Star Game. In 62 total games for Oklahoma City in 2019 he hit.268 with 20 homers and 54 RBI. Smith was recalled to the majors again on July 26 to replace Austin Barnes as the Dodgers primary catcher and the following day hit a homer and two doubles against the Washington Nationals as well as driving in six runs, the most RBI by a Dodger rookie since James Loney in 2006. With 26 RBI and nine homers in his first 23 games, he passed Cody Bellinger for the most in franchise history through that many contests. He finished the 2019 campaign by hitting.253 in 54 games with 15 homers and 42 RBI.
