Japanese starting sequence:
Get Golds on all the tracks in intermediate mode in single race mode.

Ace mode:
Select single race mode, then press L1 + R1 at the difficulty selection screen to display the "Ace level" option.

Demo mode:
Instead of allowing the game to idle until demo mode begins, press Triangle at the main menu. Each demo will play in sequence.

Toyota Pod race:
Successfully complete course license mode with a bronze or higher rank to unlock the "Pod Race" option at the game status menu.

Gran Turismo Concept 2001: Tokyo FMV sequence:
Complete 50% of the races to unlock the Gran Turismo Concept 2001: Tokyo FMV sequence at the game status screen.

Ending FMV sequence:
Complete 75% of the races to unlock the ending FMV sequence at the game status screen.

Extra credits for Gran Turismo 3 A-spec:
Win all races in single race mode under the normal difficulty setting to unlock an option at the game status menu to give 10,000,000 credits and a Bronze trophy for all licenses not already unlocked to a save file for Gran Turismo 3.

Hint: Getting silver or gold on all course licenses:
Choose "Yes" prompted to use the pace car just before you start your course license test. It helps you get at least a silver trophy, or even gold trophy on the particular test / tracks you are doing (for example, Tokyo R246). Take over the pace car and try to cross the finish/start line before the pace car does. However, on the rally course license tests/tracks, you may not be able to see where you are going due to the dust given out by the pace car. But, it still helps you to get at least a silver trophy.