It's not really important what car you choose at start, but I recommend Grinder because it has good acceleration and handling at the same time. However it's not the best car you can have.
I will use Grinder in this short picture guide showing you how I beat Woodland Havoc. I could use any of the starting levels but I managed to take good shots here.

The most important things in FlatOut racing are your driving and wrecking skills. (You also have to upgrade your car often so that you can go faster; do not forget that different types of environment are handled differently, for example in snow levels you'll have to maintain inertia and remember when to turn.)
You have to remember that in FlatOut it's not the fastest or strongest one who wins, it's always the one who comes to the finish line first. That means if you encounter an opponent you have to keep them away from you: if they have more horsepower (Grinder has less hp than say Blockhead) or are not easy to overcome you must wreck them. Do not forget that you have to keep your pace and remember that you need to come first.
When driving in mud you should not go full acceleration and only then turn because in most cases that will lead to losing speed and position. Instead you should slow down, turn patiently and only then steer. This actually applies to any kind of roads and whatnot, but to a different degree.
The better you are able to turn your car without losing much speed the more chances of winning are present.
Be careful with them.
Remember to wreck enough things so that you have a nitro boost.
As I already mentioned if your car is not the best in this race you have to be brutal.
I failed this one.
Now the fun part.
Here goes the wrecking.
One must always be careful.
Sometimes it's easy to pull off. Sometimes.
Your opponents will always keep you busy.
However if you eliminated the challengers it will buy you time.
If you made a mistake a nitro boost is handy.
They will always get you. Even if you won.

These were only the basics. Advanced gameplay comes with knowing your car's weight (the heavier it is the stronger intertia will be) and acceleration parameters. With. lots. of. practice.