In no particular order…

• Sly Cooper trilogy. The second game is the best, but the third has its strong points too. The first game feels like a tech demo in comparison to the second.
• Final Fantasy X (though I’d actually recommend playing it on PC with the Untitled Project X mod)
• Ratchet and Clank. The second and third games are the best. Deadlocked is divisive for many fans.
• Jak and Daxter trilogy. 2 and 3 are very different from the first game.
• GTA San Andreas.
• Kingdom Hearts 1 & 2. Ideally you should actually play these on the PS4/PS5/PC, because those are the directors cut (Final Mix) versions of the games. You can’t skip Chain of Memories either, which was originally only available on the GameBoy Advance; Look up a plot summary if you don’t want to play CoM, because the gameplay is very different from KH1 and 2.
• Shin Megami Tensei: Nocturne, Persona 3FES, and Persona 4. Nocturne is from the mainline SMT series, while the Persona games are a spin-off. All three are available on other systems as well, (P3FES recently got a remake, for instance).
• Devil May Cry 3: Dante’s Awakening. It’s a prequel, so you don’t need to play DMC 1 or 2 first.
• Metal Gear Solid 3: Snake Eater. It’s also a prequel, so you don’t need to play MGS 1 or 2 first. MGS Subsistence is the director’s cut version of the game, with some added gameplay improvements.
• Okami. If you like Zelda games, you’ll enjoy this. The combat is more akin to a DMC/God of War hack-n-slash, but the dungeons feel like Zelda dungeons.
• Resident Evil 4. The recent remake is also stellar.
• Shadow of the Colossus. Don’t read anything about it beforehand. Go in blind.
• Odin Sphere. A hand-drawn side scroller, with a great story.
• Viewtiful Joe. Same devs as Okami, also cell shaded but with a very different art style. Stylish, fast paced side scroller.
• Psychonauts. 3D platformer with a wild setting.
• God of War trilogy. See where the series started, because the recent games are very different while still maintaining a lot of the same.