Speed Zones are PR stunts where you have to maintain an average speed through a series of corners. As you complete Speed Zones, your Speed Zone Hero level will increase.
For paved road Speed Zones, a stock Koenigsegg Agera RS worked just fine for me. However, for off-road Speed Zones, I used an S2 Ariel Nomad.
Also, at least for me, I found that some of the Speed Zones on hills were easier to do uphill. The slope will naturally slow your car down meaning less braking is required from you.
You should get this naturally as you go for 3-starring each one on your way to the Stunt Superhero achievement.