I've only played Outrun at the arcades but to me it seemed that it was basically all about drift (think tracks designed with long fast corners and game play designed to encourage drifting); perhaps the full 'home' versions have a different emphasis.
OutRun 2 for the arcade/XBOX made ridiculous high-speed drifting the series' new bread-and-butter. OutRun 2006 is the version that was also released for PS2/PSP/PC.
There are some arcade and sim suggestions already given. For my money there are only four racers worth having on the PS2. Sims are the only way (Enthusia, ToCA3 (Pro-Sim), GT3&4).
If you make the shift to another sim-based racer go to Enthusia. You may find the swap difficult at first as drifting in Enthusia will need you to expand your knowledge of drifting. Basically the reason for this is that GT4 only allows for inertial drifting (throwing the car sideways). GT4 doesn't allow for power-oversteer so this will be an added aspect that you will have to get used to with Enthusia. If you have drifted with GT3 then you will have some power-oversteer experience already and will find the swap easier.