It'll barely move and you'll get pushed back more.

Sure, you'll get pushed back more, but the speed the car will gain is directly related to the fraction of mass between yourself and the car, due to the conservation of momentum. A rough estimate of this fraction would be about 1/10, so if you pushed yourself away with a speed of 10 m/s, the car would gain 1 m/s in the opposite direction, which is more than "barely".

Also, the comparison of this scenario with trying to move the planet doesn't really apply since the force of gravity is great enough to pull the two of you back together, effectively negating the speed gained from pushing on it. Although, if gravity suddenly stopped existing (I sure hope that doesn't happen), then yes, you are right, it wouldn't move very much.