Wedge played catcher and center field for Wichita State University from 1987–1989, leading the school to a 68–16 record and the College World Series championship in 1989.

[citation needed] Wedge's performance earned him first-team All-America honors,[3] the Missouri Valley Conference Player of the Year award[4] and he was runner-up to Ben McDonald for the Rotary Smith Award for College Baseball Player of the Year.

On October 5, 1991, he made his major league debut in his only appearance of the season for Boston with a pinch hit single off Chris George of the Milwaukee Brewers.

[9] He split catching duties with Scott Hatteberg for Pawtucket that season and made his final major-league appearances with Boston in July, going 0-for-6 in two games as a designated hitter.

[11] Wedge made his managerial debut in 1998 with the Columbus RedStixx of the South Atlantic League, leading the Cleveland single-A affiliate to an overall record of 59–81.

The Aeros finished the season 75–68, just missing the Eastern League post-season after losing a one-game playoff with Harrisburg.

[11][12] In 2001, he continued his ascent through the Indians' managerial ranks, leading their Triple-A affiliate, Buffalo Bisons, to a 91–51 first-place finish in the International League's North Division and a berth in the postseason, where they lost to Scranton/Wilkes-Barre Red Barons in the semi-finals.

Wedge was honored with his third post-season award when The Sporting News named him Minor League Manager of the Year.

[14] The 93 wins in 2005 were the eighth-most in the more than hundred-year history of the franchise and the team narrowly missed qualifying for the playoffs for the first time since 2001 when they were eliminated on the last day of the season.

In the American League Manager of the Year balloting for 2005, Wedge finished as runner-up to Ozzie Guillén.

Entering the season, they were expected to compete for the division title, but got off to a poor start and were essentially out of the race by mid-season, trailing the division-leading Detroit Tigers at the All-Star break by 18+1⁄2 games.

They finished the season in fourth place with a 78–84 record, 18 games behind the Central Division champion, Minnesota Twins.

[28] On January 23, 2025, he was announced as manager of the Tulsa Drillers, the Los Angeles Dodgers affiliate in the Double–A Texas League.

Conference tournament champion Wedge joined Baseball Tonight on ESPN as a Studio Analyst for the 2014 and 2015 seasons alongside Ozzie Guillén and Dallas Braden.