You can also get some ps1 games if you want. I think ps1 has a wider library of jrpgs. Suikoden 2, Parasite Eve, Legend of Mana, Chrono Cross, etc.
For hack and slash, I recommend Zone of the Enders 2. You can get Zone of the Enders if you want, but the 2nd blows it out of the water.
For Beat em up, get Viewtiful Joe.
Metal Gear Solid 1, 2, 3 is nice.
Get Ico and Shadow of the Colossus.
And try out Ace Combat 4, 5, 0 whenever you get the chance.
If you ever want to dabble in horror, try out Silent Hill 1, 2, 3. I haven't played 3 yet unfortunately

, but 2 will make you sad.
Katamari Damacy is good, quirky, and fun. Beyond Good and Evil is underrated and good.
If you wanna try out some fighting games, I heard Capcom vs SNK 2 is good.