I played MSR on DC a lot, I think it is a precursor to Gotham.
Sonic R on SS was good fun, but I'm in love with the music. Touring Car has good tunes also, but the game is ass.
Sega Rally on SS is my goto, feel good, relax driving game. I can play time mode for hours even though there is only 3(4) tracks.
Vanishing Point on the DC had a fun mode where you do tricks. I never liked the actual main game, just the Stunt Driver Mode.
You can't go wrong with Rage Racer. Definitely best of the lot there on PS1. Of course WipeOut should be mentioned but it's not a traditional racer.
I've got Bari Bari Densetsu on my short list, it seem like a fun obey driver. I loved hang-on for the SMS, so this should be a welcome upgrade.