I usually don't like racing games unless it's themed, colorful, has a "special" flavor to it, involves crazy physics and/or allows me to crash my cars (and gain points by doing so) or shoot missiles at my opponents, or something along the lines of "doing something else other than just following the track".
I like them when it's sort of action-oriented too (doesn't need to have weapons on the vehicles for that), has exaggerated tracks, impossible curves only drivable by a futuristic car or has some sky-crapper-high jumps and makes your car (or jet, or vehicle, whichever) race through water or magma falls for a split second that sends goosebumps in your stomach, along with some other arcade-style conditions here and there. Generally speaking I don't enjoy the likes of Grand Turismo and so on, those I find terribly boring (I refer to those as "normal" racing games).
The ones I liked over the years (decades) are...
º Mario Kart (series), particularly the N64 and Wii versions (SNES one is always classic)
º San Francisco Rush: ER (Arcades) and 2049 (Arcades or Dreamcast)
º Hydro Thunder (Arcades or Dreamcast), and H2Overdrive (Arcades)
º Wave Race 64 (water effects that still hold their ground to this day, amazing)
º Wipeout (series), original, XL and N64 versions in particular
º FlatOut 1 & 2
º Stunt Race FX (SNES), very difficult to enjoy today (runs very slowly) but was very fun back then, it's lighthearted and the music was (still is) memorable
º Star Wars: Episode I - Racer
º Extreme-G (original, specifically) and Extreme-G 3 (PS2), disliked Extreme-G 2
º Sonic & All-Stars Racing Transformed
º One of the Need for Speed on the PS1, can't recall which one but you had to race as well as escape from pursuing police cars (it's not the latest one released in 2010 or recently, that one I've seen and it's cool I guess but I'm referring to the one on the original PlayStation). I admit though I haven't played most of the NFS games in the entire series, but out of the ones I know I've played those "hot pursuit" type ones were enjoyable enough.
º Speed Devils (Dreamcast), enjoyed it a lot back then especially on-line, although I've played it again about two or three years ago and did not enjoy it as much as I used to, but still has some charm (seemed way more bland and cheesy than what I recalled, but I really did like it a lot during the Dreamcast days).
º Daytona USA (Arcades or Dreamcast), probably the sole exception of a "normal" racing game I actually enjoyed, but I think that was due to the Arcades version being so "in your face" obvious with convincing and loud audio and rumbling seats and so on, plus the visuals at the time were superb (not sure if I would still enjoy it today, haven't played it since I rented the Dreamcast version back in early 2000 or around that time)
º Sega Rally Championship and SG2 (Arcades), same as above for Daytona USA, I think I enjoyed them due to their arcades versions having an impact on my perception of racing games at the time (which, back then, I never enjoyed at all lest for the likes of Mario Kart).
And a few others I'm sure I'm not thinking of right now but yeah, you get the idea of the type of racing games I like. And out of those I mentioned I would easily place the Mario Kart series first on the podium (not one in particular, just all of them, but I guess I'd pick the Wii version since it's the latest I've really enjoyed).