The first touchdown scored in NFL history occurred on September 26th, 1920 in a game between the Rock Island Independents and the St. Paul Ideals. The league at the time was called the APFA and the Independents were a member. Independent's running back Eddie Novak scored the first ever touchdown on a 10 yard rush. The Independents would win the game 48-0 and Eddie Novak would score two touchdowns on the day.
However, the St. Paul Ideals were not a member of the APFA so there is some disagreement to what actually counts as the first NFL touchdown. The first game between APFA opponents came on October 3, 1920 between the Dayton Triangles and the Columbus Panhandles. Triangle's player Lou Partlow scored the first touchdown in that game. The Dayton Triangles would win that day 14-0.
In 1922, the APFA renamed themselves the National Football League or NFL as it is known today. The first touchdown in the newly named league came on October 1st though it is unclear as to which player scored it as several games were occurring simultaneously.
In 1966, the NFL decided to merge with the recently created AFL, with it becoming effective in 1970; The first touchdown was scored in the first game of the season between the St Louis Cardinals and Los Angeles Rams. Willie Ellison caught a three yard pass from Roman Gabriel on September 18th.
The first touchdown in the Super Bowl as scored by Max Mcgee on a 37 yard catch and run in the first ever Super Bowl. Mcgee was a backup for the Green Bay Packers who came in due to an injury. The Packers ended up beating the Kansas City Chiefs 35-10 to secure the first ever Super Bowl.