A few more memories that stand out:
Mid 2002, I went to a cousin's house that lives quite a ways away, and stayed up ten or so hours playing Final Fantasy IX on his new PS2. The charming characters, art direction, music and sheer nostalgia of older Final Fantasy elements had my eyes glued to the screen the entire time.
During that same time, the Dreamcast was big, so I happened to come across Resident Evil: Code Veronica. Many of the game's puzzles were confusing and nearly impossible to me, but again and again I delved into the mysteries of the isolated prison facility. I also spent a great deal of time getting my ass kicked by Tyrant 2 before figuring out the secret to his defeat.