1988 was probably the first year where I was old enough to have a basic grasp of sports. My family had just moved to a Los Angeles suburb and it was fortuitous timing with the Lakers and Dodgers winning that year. Gretzky would be traded to LA in August '88 but hockey was far down the totem pole at that point.
The Kings knocking Calgary out in 1990 was the first NHL game I could remember actively caring about. As I remember it, we were listening on the radio during a long car ride and got home just in time to see the OT goal.
The Kings 1993 run probably was the main catalyst in hockey grabbing more of my interest with baseball, basketball, and football. Mix in the 1994 MLB strike, the Rams/Raiders leaving after the 1994 NFL season, and the Lakers not contending, suddenly the NHL became the new hotness (despite the 1994-95 lockout).
I switched schools in Fall 1993 and all of the "cool" kids were into hockey. At that point I only knew the Kings roster and the cool kids were talking up this Pavel Bure guy. I had to beg for NHLPA '93 for the SNES to learn about the other teams (simpler days before the Internet).
When I got NHL '95 for PC, I now had the ability to add prospects to the game. I got a little OCD about that and started following the draft from that point on.
I forget if the short-lived 1991 Pro Stars cartoon had any affect on me. Ie, the cartoon was telling me that Gretzky was on par with Michael Jordan and Bo Jackson, so I ought to watch.
I hate to admit it, but The Mighty Ducks (1992) and its sequel (1994) probably helped further establish hockey for me.