My favorite one has always been on the Gamecube. Probably because it's the only one that lets you pick 2 characters for one kart, and it's the first one I played.
I played the Wii and the DS one, too, but I don't like those as much as the Gamecube one.
The console that I enjoyed Mario Kart the most on is the Nintendo 64. I think I played it on the GameCube as well, and it was very fun. I can't go into detail about it, since I don't remember much from those days except that I really liked those games a lot back then.
Mariokart Double Dash was the best, you can have co-op and the tracks are very good, along with having special items per character and different karts based on what characters you pick. The Wii version is lame, it's very easy and the tracks aren't all that great, and the DS versions are all the exact same. The N64 version was good, but there weren't all that many places.
I like MK64 the best! It was the hardest, but somehow I got all gold. I like that some of the tracks were on Wii. I've got all gold on MKDD and MKW too, but I haven't played handheld or NES or SNES.
Well, I'd have to say Wii, because you can play with your friends and the tracks are larger and cooler. However, N64 MarioKart has that nice, retro feel...
I've played it on the Wii, Nintendo (DS/i/XL), and GameCube.
1. GameCube (I love this version) 2. Wii (It's like actually driving if you have the wheel controller) 3. Nintendo DS/i/XL (Not as entertaining, gets boring quick without other real players)