The PS2 is going to be the best 'catch all' for gaming needs, as it has the highest installed user base and thus far and away the most titles. However, it is also far and away the most poorly manufactured console, as is par for the course for Sony. And for it's massive library, there are only a handful of must-play titles, and fewer multiplayer titles still.
The Gamecube on the other hand, lacks the library of the PS2 in terms of quantity, but the quality of its exclusives makes up for it. It also features more titles with content appropriate for all ages, and is the best made and cheapest console out there. For what you seem to be after, I would recommend a Gamecube to you first if you came to my shop, and titles like F-Zero GX (racing), Super Monkey Ball (insanity), Mario Kart, Zelda and possibly Metroid: Prime. None of these games would cost you over $20 bucks, either.
The Xbox I would only recommend here because of the multiplayer and racing games. However, the content here tends to be of the more mature variety, or simulationist racing, and really about Xbox Live.