In Death Rally 2012 the beginning of the game can be quite tough. From your basic Vagabond start races and stay behind (let everyone overtake you from the starting line), and simply chase the 5th car, without overtaking, but try to kill him. Run only the pit, where avenues are long and allow even the basic gun to be very effective. Make picking up parts your goal, this is really key to progressing. During the first 10 races you should not try to win, simply kill the guy in front and picking up parts.
Spare nitros to chase number 4 once you have killed number 5 (or to catch up when you are too far behind). One last nitro at the right moment in the 3rd lap can make a big difference and get you 2nd or even 1st with an inferior car. Remember: unused nitros are lost after the race.
Invest all on speed, and then on the weapon. This strategy will provide you lots of parts which will allow you to move on to a faster car and weapons very quickly.
Forget about the other tracks or one-on-one until you have a flamer and a wraith: you will earn more on the pit by killing one, then two, then three foes than by running more complex (curvy) tracks where your weapons and cars will be too weak to really pay off.
This may sound a bit dull but the goal of the game is to advance: fame will follow fast.
In particular this strategy allows you to suffer very little damage (there is no one behind you to cause you damage) and therefore invest a maximal share of your income in speed/weapon upgrades. When you get mines, do not try to use them offensively (when overtaking) but drop all three in a "wall" (across the track) as this will often provide an easy kill on leading adversaries who may already be weak.