Weinke was born and raised in St. Paul, Minnesota, where he attended Cretin-Derham Hall High School and was a three-sport star, playing first base for the baseball team, quarterback for the football team, and was captain of the hockey team. In 1989 during his senior year in high school, he was a Parade magazine and USA Today first team All-America selection, was named Minnesota's prep football player of the year, and was seen as the top senior quarterback in the country. Weinke was recruited by over 70 Division 1 schools, including Alabama, Arizona, Arizona State, Florida State, Illinois, Minnesota, Miami, Washington, and Wisconsin, but ultimately signed a national letter of intent and committed to play quarterback for the Florida State Seminoles of Florida State University despite being a diehard Miami Hurricanes fan (due to fellow Cretin-Derham Hall alumnus Steve Walsh attending the school and starring for the team at quarterback). However, he was also an all-state baseball player and was drafted in the 2nd Round of the 1990 Major League Baseball First-Year Player draft (the 62nd player taken overall) by the Toronto Blue Jays. After spending four days in August 1990 on the FSU campus, Weinke put his college career on hold and instead signed a contract to play professional baseball and reported to the Toronto Blue Jays minor league baseball system. Head football coach Bobby Bowden promised Weinke that he would always have a scholarship offer if he wished to return.