I'll be mentioning Gamepro magazines a lot since many of the games I now know and love were because of Gamepro or any instances were i read an issue of it.
Shin Megami Tensei:
I'm a little late on this one but SMT: Nocturne was the SMT game that got my attention since one of my friends had it.
Final Fantasy (and many many PSone RPGS):
Although our family friend has FF7 and it was the first FF game i saw, I never had the chance to play it since I couldn't find legit copies of it. So when FF8 came out, thats what got me to the whole FF Craze.
Command and Conquer:
Red Alert got my attention. Mammoth Tanks.
Persona:
I never really played or had anyone having P1 or P2 with them so my first persona game was 3. A college friend of mine had P3 on his PS2 and I was amazed at the quality voice acting of the English cast. Also, Yukari... yep he had to be playing on some scenes were the P3 MC was interacting with Yukari and i immediately said - DONE. ATLUS TAKE MY MONEY! Now i got hooked with Persona, so i bought my own copy of P3 from a friend who was already done with it, P4, and now waiting for P5.
Street Fighter and other fighting games
I remember back in the day, I would go to a bowling alley with my mom and some cousins in order to have some fun outside of home. I didn't like bowling so i went to the arcades with the cousins and they had a Super Street Fighter 2 Turbo cabinet and that's what got me, along with Marvel Super Heroes (got me into the Marvel vs Series), Darkstalkers, and Tekken 2. My older cousin was already good at all of those games since he spent a lot of time in arcades and he helped me how to properly handle playing on arcade sticks and poof, Street Fighter became my favorite fighting game for a time.
Guilty Gear
It was years ago when people were too busy playing their PSX console ports of Street Fighter and Tekken, I needed to find a game that I would actually like better than any of those series. So I was reading an issue of Gamepro (i think) and saw this ad for Guilty Gear: The Missing Link aka the first Guilty Gear Game. I was skeptical at first since I didn't like the game's actual graphics based from the screenshots but the artwork and the character designs look way cooler and edgier than SF and Tekken (sorry kiddos, /shotgunsfired). I never found anyone having the game nor a store selling it, so i was sad and unhappy. Then years later, Guilty Gear X would come out and had, for its time and even for today, amazing 2d in-game graphics and i got sold. I saw it first on an issue of Gamepro and i knew i had to play the game. Lucky for me, i found a mall's arcade rink where they had a GGX cabinet and unlucky for those other people who wanted to play this "not tekken or sf game", i kinda wasted them with Sol thanks to that gamepro issue. So yeah, the much better graphics of Guilty Gear X (and the rest of the GG series) helped me establish a love for the said series. I also forgot to mention, Guilty Gear's music is amazing.
Spyro the Dragon:
Remember those good ol days in the PSone where we got Demo discs! Yep, i remember the first Spyro game being part of a demo disc and I immediately fell in love with it. It was a vastly different experience from playing Crash Bandicoot since Crash's gameplay was not the "3d free roaming adventure like Mario 64" that we hoped to be. Ever since playing the Spyro demo on that demo disc, we finally got a copy of Spyro 1 and spent restless nights finishing as many levels and collecting as many gems as we can.