Mine was Super Mario RPG. At the time, I was most definitely not an RPG fan, but I was a Mario fan, so when I saw it up for rent, I knew I had to try it. And, boy, did I have no earthly clue how the hell I was supposed to play it.
Well, that's not entirely true. I kinda was able to wing it, until I got to the Hammer Bros. At that point, I promptly lost. I tried many times, and lost to what was arguably the easiest boss battle in the game (and certainly the first one, as Bowser didn't count for anything ). I returned the game, feeling a bit disappointed (I think, it was an eternity ago, really!)
Much later on, I decided to give that game another chance, especially since I had a dream telling me to do so (yes, you heard that right. A dream told me to give Super Mario RPG another go-around. Though, to be fair, it was a pretty standard dream, not like those BS lines of dialogue in Paper Mario where characters talk about oracles visiting in their dreams, but my dream's message to me was clear all the same). Well, at that point, I couldn't rent any Super NES games anymore, as we were already well into the 5th generation (32/64 bit era), so I had to buy it used. Naturally, I did just that, and found out I couldn't stop playing it. Well, that's not really how it was, as I got stuck at times, but I loved it all the same. I think it was at that point I realized I liked RPGs. Though I don't really remember when it was, actually, now that I think about it, but I think that was it.
Well, in any case, I can definitely say SMRPG was the first game of the genre I played.
Oh, and yes, I beat that game after a few years (Pathetic, huh?). Currently playing through a hard type hack of it called SMRPG Remix, as I consider myself fairly knowledgeable of the ins and outs of that game, but...yeah.