For example, I used to not like JRPGs. Something about the setting just put me off. But then one day I tried Chronotrigger out of simple boredom and was quickly hooked. And just like that I developed a taste for
all the big players in the genre that I dismissed before, like all earlier Final Fantasy's.

Similarly Fallout made me realise that rpg games that [bi]switch over to turn-based for combat[/i] can be really good. Before I simply stopped reading a game review/preview once I got to that phrase.

As for films, I can't really think of an example. The closest I've got is the, uhm, genre of Tom Cruise films. The dude's face and life annoys me to the extent that I don't want to see any of his films. But then I see something like Edge of Tomorrow and can't help but to admit that I enjoyed the hell out of it.

Any similar examples?

Wasteland got me into post-apoc genre.
Blade Runner the movie got me into cyberpunk or some sort of prototype version of cyberpunk.

"Duke 3D" - FPS

"StarCraft" - RTS

"I Love You Traci" - Porno

"Tauto" - Putting lit cigarettes out on my crotch.

Jamestown is the first bullet hell shmup I've played. (I don't really play this genre much, but I like watching videos, particularly high score videos, of such games).

Pokemon Mystery Dungeon, and later, Shiren the Wanderer DS, sort of got me into roguelikes.

StarCraft and WarCraft II are the only two RTS I can tolerate to this day. I want to love AoE and the like, or Command and Conquer... But I can't I played StarCraft and WarCraft II both all the way through the campaign modes and I LOVED it.

Devil May Cry got me into a seriously fun Action adventure genre that includes excellent gems such as Bayonetta, God of War, Darksiders and Onimusha. Before I played the first DMC, action adventure was only Zelda for me.

My first horror film was a Nightmare on Elm Street. I have since loved horror flicks, and even gotten into some that weren't my speed before such as Alien and Halloween.

We (me and the misses) quite enjoy the Sub Genre of "Tom Cruise" movies, which is "Painful things happen to Tom Cruise"

Any film where he dies repeatedly is high up on that list.
All the M.I. films has Tom face planting one way or another.

I think as an Actor he is disturbingly generic, Rainman is the only film where the quality of the film would not have been lowered had any other actor done the role. The ball on a stick that was the stand-in for Jar Jar binks during filming could easily replace him in most of his films.

Pride and prejudice was the first pre 1900 costume drama I enjoyed.

I liked the Last Samurai, and instantly thought of 4 other actors (one who was actually in the film) that would have pulled the role off better. Tom Cruise gets roles in good movies where almost anyone could look good in the role.

Half Life.

FPS were one of my favourite genres since I liked Wolfenstein, but by the end of the 90s I was really tired of them and almost gave up. Half Life caused me to start playing them again.

(So, in this case it would be a game that broke me into an "old" genre)

Ah yes roguelikes, as in genuine roguelikes similar to rogue. Again, simple boredom had me trying my hand at Nethack one day, and I could only wrestle myself away from the game 2 or 3 days later. Soon after, I wanted to check out ADOM and similar games as well. Nethack ended up doing more than just developing a taste for roguelikes though. It made me curious about other ascii games as well. Dwarf Fortress and Cataclysm : DDA specifically have by now eaten up several weeks of my time :P

Operation Flashpoint: Cold War Crisis got me into infantry/realism/authenticity simulators.

victor vran and torchlight

never opted to play those action rpgs before but then i realized its a nice mindless way to blow off steam for an hr or two, plus they are easy to set aside for weeks, months etc and just jump back into play when you get the itch