Haha, that is a long time. The first time I played Daytona was a year or two before the Saturn version, in the Smokey arcades of Broadway NY. Now we have no arcades in Broadway. :(
Manual will actually help a scub, all you have to do is just to learn the basics. :)
Use the 4 face buttons on your Dreamcast controller. Gear 1 Yellow button, gear 2 red button, gear 3 green button, gear 4 blue button, left trigger for brakes and right trigger for the accelerator. When you first start to drive, you don't need to hit gear 1, just press the accelerator, when it's time gear up, then press gear 2, then gear 3 and finally gear 4. During the tough turns, press gear 1, then press gear 3 to drift, no brakes needed. Which gear you want to use during a turn, is all up to you, for beginners I recommend gear 2 then gear 4 or 1,3 and 4. You can tap the brake during some of the tough drifts, that should help you a little. Start with the Grasshopper, then once you master the manual controls, then pick the car of your choice, or stick with Grasshopper which is the easiest car to drive.
The first few minutes might feel a little complicated, but after 10 or 15 minutes, you will see how easy is to drive, once you master the manual controls, you will never touch another wall or the grass during any drift.
Give it a try, I hope you master it. :)