Then you missed out on the greatest system ever.  Play DRAGON QUEST VIII, it's so goddamn good, and so goddamn long.  Also play both Kingdom Hearts, Timesplitters 2, Jak and Daxter series, Sly Cooper series, the Warriors, so many others

Fatal Frame series
RE Code Veronica (if you never played it on Dream Cast)

Kingdom Hearts is a must! I personally love going back to SSX: Tricky and 3 recently and MGS obviously :D

get metal gear solid 2 and 3, great games.
 
get god of war collection 
 
also get shadow of collosus when that game comes out.

I feel sorry for any gamer who never owned a PS2. You certainly have a LOT of catching up to do. I would start with:

The Jak Series

Ratchet and Clank series

God of War 1 & 2

Cold Winter (one of the best FPS for consoles)

Silent Hill 2 & 3

ObsCure 1 & 2

Onimusha series

Devil May Cry series (you could probably skip 2, though)

Resident Evil: CVX

These are just off the top of my head. There are dozens of PS2 games worth playing.

These are my favorite PS2 games, not in any specific order:

1. Kingdom Hearts 1/2
2. Ratchet and Clank (any of them)
3. GTA San Andreas
4. Burnout Revenge
5. Devil May Cry (haven't played 2 or 3)
6. Shadow of the Colossus
7. Onimusha 2 (only one that I played)
8. NBA Street 2
9. NFL Street 1/2
10. Street Fighter EX2

For my part I would recommend;

• Jak and Daxter 1, 2 & 3
• Okami
• Ratchet and Clank 1, 2 & 3
• MGS 2 & 3
• SSX: Tricky & 3
• GT4
• Viewtiful Joe

I wouldn't recommend playing any shooters for the most part since they really don't age very well. Also I would have included GoW games and Ico/SOTC but you should really just pick up the collections for your PS3.

Wait, did you ever own a PS1?

So many games..........

I really liked:

1. Metal Arms: A Glitch in the System
2. The Mark of Kri
   
3. Dark Cloud 2
4. Fatal Frame 2: Crimson Butterfly
5. Burnout 3: Takedown

Onimusha series

Also Devil May Cry, Devil May Cry 3 and most of the Rachet and Clanks

@Wolverine said:

When I was younger I had a Xbox and Gamecube, but never a PS2.

I bet those words make a lot of people here feel old. I can barely accept people growing up with PS1 being their first console, let alone PS2...

I am disappointed in so many of you because of your recommendations. Now that I've said that, I can't recommend games myself without them being ripped apart, so... bummer.

Most of what I would suggest has already been suggested. Jak and Daxter should also be looked into. Okami and Dragon Quest VIII are obscure great games too.
I can't comprehend the amount of games I would recommend.

I'd recommend the following exclusives: 

• God Hand
• The King of Fighters XI
• Zone of the Enders: The 2nd Runner
• Devil May Cry
• Devil May Cry 3
• Project Zero 3: The Tormented

You're in for the long run now...

Psi-Ops, Metal Gear Solid 2 and 3, Prince of Persia Trilogy to name a few.

Kingdom Hearts is a must! I personally love going back to SSX: Tricky and 3 recently and MGS obviously :D

God of War collection, GET IT NOW. the best remastered games. 
 
Anyway yeah Final Fantasy X, XII, and Kingdom Hearts 1 & II all great games. 
 
MEtal Gear Solid 3 is also being remastered although i hope to god it's Subsistence.

• Persona 3 FES
• Persona 4
• Rogue Galaxy
• Disgaea: Hour of Darkness
• Disgaea 2: Cursed Memories
  
• Eternal Poison
  

When I was younger I had a Xbox and Gamecube, but never a PS2. I missed out on so many great games and I wanted one so badly. Relatively recently I bought a PS2 from my friend and I'm starting to go through old games that I missed out on like Final Fantasy X and Kingdom Hearts. What recommendations do you have?

Also, I have a PS3. Should I wait to play games like Shadow of the Colossus until they are remastered?