This site lists all of the goals a player ever scored for the Flyers:
http://www.flyershistory.com/cgi-bin/goalscorerlist.cgi
Obviously, if they didn't start their career there, it wouldn't be the first NHL goal.
For the remainder of the teams/players, the NHL has *FINALLY* completed game logs for every NHL player.
https://www.nhl.com/player
You can search any player and see their game log, which will let you figure out when their first game, goal, point, etc. was.
It used to be that this information only went back to 1987-88, so a pretty cool new feature. For example, here's Gordie Howe's game log from his first season:
LINK