It's extremely easy to complete this mission with any low-rider as long as you don't crash into the other cars or traffic. Most low-riders spin out of control quite easily on impact with other objects. When the race starts, take the right or left footpath to avoid the cars in front of you. You might get slowed down due to running over pedestrians but you can still grab the lead at the first turn itself since the other racers will crash almost all the time. For there on, there's not much traffic, the road is wide and the race becomes piss easy.
The roads are a suggestion, drive as straight as you can and there's no need to follow the road if any other path is available. Don't hit (on purpose) the other competitors as it will make them angry. The other drivers in the race(s) will get out of your way if you play nice. The Red Pylons are the only required to make you follow a path. You also don't have to drive through them, just come close (experience will show you how much leeway you have). Any acceptable vehicle is good for any race (if it isn't raining), though I'm suggesting you to do the race with either Remington or Voodoo.